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In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma. Roughly speaking, the theory asserts that given any sequence ${G_n = (V_n, E_n)}$ of finite graphs, one can extract a subsequence ${G_{n_j} = (V_{n_j}, E_{n_j})}$ which converges (in a specific sense) to a continuous object known as a “graphon” – a symmetric measurable function ${p\colon [0,1] \times [0,1] \rightarrow [0,1]}$. What “converges” means in this context is that subgraph densities converge to the associated integrals of the graphon ${p}$. For instance, the edge density

$\displaystyle \frac{1}{|V_{n_j}|^2} |E_{n_j}|$

converge to the integral

$\displaystyle \int_0^1 \int_0^1 p(x,y)\ dx dy,$

the triangle density

$\displaystyle \frac{1}{|V_{n_j}|^3} \lvert \{ (v_1,v_2,v_3) \in V_{n_j}^3: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_1\} \in E_{n_j} \} \rvert$

converges to the integral

$\displaystyle \int_0^1 \int_0^1 \int_0^1 p(x_1,x_2) p(x_2,x_3) p(x_3,x_1)\ dx_1 dx_2 dx_3,$

the four-cycle density

$\displaystyle \frac{1}{|V_{n_j}|^4} \lvert \{ (v_1,v_2,v_3,v_4) \in V_{n_j}^4: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_4\}, \{v_4,v_1\} \in E_{n_j} \} \rvert$

converges to the integral

$\displaystyle \int_0^1 \int_0^1 \int_0^1 \int_0^1 p(x_1,x_2) p(x_2,x_3) p(x_3,x_4) p(x_4,x_1)\ dx_1 dx_2 dx_3 dx_4,$

and so forth. One can use graph limits to prove many results in graph theory that were traditionally proven using the regularity lemma, such as the triangle removal lemma, and can also reduce many asymptotic graph theory problems to continuous problems involving multilinear integrals (although the latter problems are not necessarily easy to solve!). See this text of Lovasz for a detailed study of graph limits and their applications.

One can also express graph limits (and more generally hypergraph limits) in the language of nonstandard analysis (or of ultraproducts); see for instance this paper of Elek and Szegedy, Section 6 of this previous blog post, or this paper of Towsner. (In this post we assume some familiarity with nonstandard analysis, as reviewed for instance in the previous blog post.) Here, one starts as before with a sequence ${G_n = (V_n,E_n)}$ of finite graphs, and then takes an ultraproduct (with respect to some arbitrarily chosen non-principal ultrafilter ${\alpha \in\beta {\bf N} \backslash {\bf N}}$) to obtain a nonstandard graph ${G_\alpha = (V_\alpha,E_\alpha)}$, where ${V_\alpha = \prod_{n\rightarrow \alpha} V_n}$ is the ultraproduct of the ${V_n}$, and similarly for the ${E_\alpha}$. The set ${E_\alpha}$ can then be viewed as a symmetric subset of ${V_\alpha \times V_\alpha}$ which is measurable with respect to the Loeb ${\sigma}$-algebra ${{\mathcal L}_{V_\alpha \times V_\alpha}}$ of the product ${V_\alpha \times V_\alpha}$ (see this previous blog post for the construction of Loeb measure). A crucial point is that this ${\sigma}$-algebra is larger than the product ${{\mathcal L}_{V_\alpha} \times {\mathcal L}_{V_\alpha}}$ of the Loeb ${\sigma}$-algebra of the individual vertex set ${V_\alpha}$. This leads to a decomposition

$\displaystyle 1_{E_\alpha} = p + e$

where the “graphon” ${p}$ is the orthogonal projection of ${1_{E_\alpha}}$ onto ${L^2( {\mathcal L}_{V_\alpha} \times {\mathcal L}_{V_\alpha} )}$, and the “regular error” ${e}$ is orthogonal to all product sets ${A \times B}$ for ${A, B \in {\mathcal L}_{V_\alpha}}$. The graphon ${p\colon V_\alpha \times V_\alpha \rightarrow [0,1]}$ then captures the statistics of the nonstandard graph ${G_\alpha}$, in exact analogy with the more traditional graph limits: for instance, the edge density

$\displaystyle \hbox{st} \frac{1}{|V_\alpha|^2} |E_\alpha|$

(or equivalently, the limit of the ${\frac{1}{|V_n|^2} |E_n|}$ along the ultrafilter ${\alpha}$) is equal to the integral

$\displaystyle \int_{V_\alpha} \int_{V_\alpha} p(x,y)\ d\mu_{V_\alpha}(x) d\mu_{V_\alpha}(y)$

where ${d\mu_V}$ denotes Loeb measure on a nonstandard finite set ${V}$; the triangle density

$\displaystyle \hbox{st} \frac{1}{|V_\alpha|^3} \lvert \{ (v_1,v_2,v_3) \in V_\alpha^3: \{v_1,v_2\}, \{v_2,v_3\}, \{v_3,v_1\} \in E_\alpha \} \rvert$

(or equivalently, the limit along ${\alpha}$ of the triangle densities of ${E_n}$) is equal to the integral

$\displaystyle \int_{V_\alpha} \int_{V_\alpha} \int_{V_\alpha} p(x_1,x_2) p(x_2,x_3) p(x_3,x_1)\ d\mu_{V_\alpha}(x_1) d\mu_{V_\alpha}(x_2) d\mu_{V_\alpha}(x_3),$

and so forth. Note that with this construction, the graphon ${p}$ is living on the Cartesian square of an abstract probability space ${V_\alpha}$, which is likely to be inseparable; but it is possible to cut down the Loeb ${\sigma}$-algebra on ${V_\alpha}$ to minimal countable ${\sigma}$-algebra for which ${p}$ remains measurable (up to null sets), and then one can identify ${V_\alpha}$ with ${[0,1]}$, bringing this construction of a graphon in line with the traditional notion of a graphon. (See Remark 5 of this previous blog post for more discussion of this point.)

Additive combinatorics, which studies things like the additive structure of finite subsets ${A}$ of an abelian group ${G = (G,+)}$, has many analogies and connections with asymptotic graph theory; in particular, there is the arithmetic regularity lemma of Green which is analogous to the graph regularity lemma of Szemerédi. (There is also a higher order arithmetic regularity lemma analogous to hypergraph regularity lemmas, but this is not the focus of the discussion here.) Given this, it is natural to suspect that there is a theory of “additive limits” for large additive sets of bounded doubling, analogous to the theory of graph limits for large dense graphs. The purpose of this post is to record a candidate for such an additive limit. This limit can be used as a substitute for the arithmetic regularity lemma in certain results in additive combinatorics, at least if one is willing to settle for qualitative results rather than quantitative ones; I give a few examples of this below the fold.

It seems that to allow for the most flexible and powerful manifestation of this theory, it is convenient to use the nonstandard formulation (among other things, it allows for full use of the transfer principle, whereas a more traditional limit formulation would only allow for a transfer of those quantities continuous with respect to the notion of convergence). Here, the analogue of a nonstandard graph is an ultra approximate group ${A_\alpha}$ in a nonstandard group ${G_\alpha = \prod_{n \rightarrow \alpha} G_n}$, defined as the ultraproduct of finite ${K}$-approximate groups ${A_n \subset G_n}$ for some standard ${K}$. (A ${K}$-approximate group ${A_n}$ is a symmetric set containing the origin such that ${A_n+A_n}$ can be covered by ${K}$ or fewer translates of ${A_n}$.) We then let ${O(A_\alpha)}$ be the external subgroup of ${G_\alpha}$ generated by ${A_\alpha}$; equivalently, ${A_\alpha}$ is the union of ${A_\alpha^m}$ over all standard ${m}$. This space has a Loeb measure ${\mu_{O(A_\alpha)}}$, defined by setting

$\displaystyle \mu_{O(A_\alpha)}(E_\alpha) := \hbox{st} \frac{|E_\alpha|}{|A_\alpha|}$

whenever ${E_\alpha}$ is an internal subset of ${A_\alpha^m}$ for any standard ${m}$, and extended to a countably additive measure; the arguments in Section 6 of this previous blog post can be easily modified to give a construction of this measure.

The Loeb measure ${\mu_{O(A_\alpha)}}$ is a translation invariant measure on ${O(A_{\alpha})}$, normalised so that ${A_\alpha}$ has Loeb measure one. As such, one should think of ${O(A_\alpha)}$ as being analogous to a locally compact abelian group equipped with a Haar measure. It should be noted though that ${O(A_\alpha)}$ is not actually a locally compact group with Haar measure, for two reasons:

• There is not an obvious topology on ${O(A_\alpha)}$ that makes it simultaneously locally compact, Hausdorff, and ${\sigma}$-compact. (One can get one or two out of three without difficulty, though.)
• The addition operation ${+\colon O(A_\alpha) \times O(A_\alpha) \rightarrow O(A_\alpha)}$ is not measurable from the product Loeb algebra ${{\mathcal L}_{O(A_\alpha)} \times {\mathcal L}_{O(A_\alpha)}}$ to ${{\mathcal L}_{O(\alpha)}}$. Instead, it is measurable from the coarser Loeb algebra ${{\mathcal L}_{O(A_\alpha) \times O(A_\alpha)}}$ to ${{\mathcal L}_{O(\alpha)}}$ (compare with the analogous situation for nonstandard graphs).

Nevertheless, the analogy is a useful guide for the arguments that follow.

Let ${L(O(A_\alpha))}$ denote the space of bounded Loeb measurable functions ${f\colon O(A_\alpha) \rightarrow {\bf C}}$ (modulo almost everywhere equivalence) that are supported on ${A_\alpha^m}$ for some standard ${m}$; this is a complex algebra with respect to pointwise multiplication. There is also a convolution operation ${\star\colon L(O(A_\alpha)) \times L(O(A_\alpha)) \rightarrow L(O(A_\alpha))}$, defined by setting

$\displaystyle \hbox{st} f \star \hbox{st} g(x) := \hbox{st} \frac{1}{|A_\alpha|} \sum_{y \in A_\alpha^m} f(y) g(x-y)$

whenever ${f\colon A_\alpha^m \rightarrow {}^* {\bf C}}$, ${g\colon A_\alpha^l \rightarrow {}^* {\bf C}}$ are bounded nonstandard functions (extended by zero to all of ${O(A_\alpha)}$), and then extending to arbitrary elements of ${L(O(A_\alpha))}$ by density. Equivalently, ${f \star g}$ is the pushforward of the ${{\mathcal L}_{O(A_\alpha) \times O(A_\alpha)}}$-measurable function ${(x,y) \mapsto f(x) g(y)}$ under the map ${(x,y) \mapsto x+y}$.

The basic structural theorem is then as follows.

Theorem 1 (Kronecker factor) Let ${A_\alpha}$ be an ultra approximate group. Then there exists a (standard) locally compact abelian group ${G}$ of the form

$\displaystyle G = {\bf R}^d \times {\bf Z}^m \times T$

for some standard ${d,m}$ and some compact abelian group ${T}$, equipped with a Haar measure ${\mu_G}$ and a measurable homomorphism ${\pi\colon O(A_\alpha) \rightarrow G}$ (using the Loeb ${\sigma}$-algebra on ${O(A_\alpha)}$ and the Baire ${\sigma}$-algebra on ${G}$), with the following properties:

• (i) ${\pi}$ has dense image, and ${\mu_G}$ is the pushforward of Loeb measure ${\mu_{O(A_\alpha)}}$ by ${\pi}$.
• (ii) There exists sets ${\{0\} \subset U_0 \subset K_0 \subset G}$ with ${U_0}$ open and ${K_0}$ compact, such that

$\displaystyle \pi^{-1}(U_0) \subset 4A_\alpha \subset \pi^{-1}(K_0). \ \ \ \ \ (1)$

• (iii) Whenever ${K \subset U \subset G}$ with ${K}$ compact and ${U}$ open, there exists a nonstandard finite set ${B}$ such that

$\displaystyle \pi^{-1}(K) \subset B \subset \pi^{-1}(U). \ \ \ \ \ (2)$

• (iv) If ${f, g \in L}$, then we have the convolution formula

$\displaystyle f \star g = \pi^*( (\pi_* f) \star (\pi_* g) ) \ \ \ \ \ (3)$

where ${\pi_* f,\pi_* g}$ are the pushforwards of ${f,g}$ to ${L^2(G, \mu_G)}$, the convolution ${\star}$ on the right-hand side is convolution using ${\mu_G}$, and ${\pi^*}$ is the pullback map from ${L^2(G,\mu_G)}$ to ${L^2(O(A_\alpha), \mu_{O(A_\alpha)})}$. In particular, if ${\pi_* f = 0}$, then ${f*g=0}$ for all ${g \in L}$.

One can view the locally compact abelian group ${G}$ as a “model “or “Kronecker factor” for the ultra approximate group ${A_\alpha}$ (in close analogy with the Kronecker factor from ergodic theory). In the case that ${A_\alpha}$ is a genuine nonstandard finite group rather than an ultra approximate group, the non-compact components ${{\bf R}^d \times {\bf Z}^m}$ of the Kronecker group ${G}$ are trivial, and this theorem was implicitly established by Szegedy. The compact group ${T}$ is quite large, and in particular is likely to be inseparable; but as with the case of graphons, when one is only studying at most countably many functions ${f}$, one can cut down the size of this group to be separable (or equivalently, second countable or metrisable) if desired, so one often works with a “reduced Kronecker factor” which is a quotient of the full Kronecker factor ${G}$. Once one is in the separable case, the Baire sigma algebra is identical with the more familiar Borel sigma algebra.

Given any sequence of uniformly bounded functions ${f_n\colon A_n^m \rightarrow {\bf C}}$ for some fixed ${m}$, we can view the function ${f \in L}$ defined by

$\displaystyle f := \pi_* \hbox{st} \lim_{n \rightarrow \alpha} f_n \ \ \ \ \ (4)$

as an “additive limit” of the ${f_n}$, in much the same way that graphons ${p\colon V_\alpha \times V_\alpha \rightarrow [0,1]}$ are limits of the indicator functions ${1_{E_n}\colon V_n \times V_n \rightarrow \{0,1\}}$. The additive limits capture some of the statistics of the ${f_n}$, for instance the normalised means

$\displaystyle \frac{1}{|A_n|} \sum_{x \in A_n^m} f_n(x)$

converge (along the ultrafilter ${\alpha}$) to the mean

$\displaystyle \int_G f(x)\ d\mu_G(x),$

and for three sequences ${f_n,g_n,h_n\colon A_n^m \rightarrow {\bf C}}$ of functions, the normalised correlation

$\displaystyle \frac{1}{|A_n|^2} \sum_{x,y \in A_n^m} f_n(x) g_n(y) h_n(x+y)$

converges along ${\alpha}$ to the correlation

$\displaystyle \int_G \int_G f(x) g(y) h(x+y)\ d\mu_G(x) d\mu_G(y),$

the normalised ${U^2}$ Gowers norm

$\displaystyle ( \frac{1}{|A_n|^3} \sum_{x,y,z,w \in A_n^m: x+w=y+z} f_n(x) \overline{f_n(y)} \overline{f_n(z)} f_n(w))^{1/4}$

converges along ${\alpha}$ to the ${U^2}$ Gowers norm

$\displaystyle ( \int_{G \times G \times G} f(x) \overline{f(y)} \overline{f(z)} f_n(x+y-z)\ d\mu_G(x) d\mu_G(y) d\mu_G(z))^{1/4}$

and so forth. We caution however that some correlations that involve evaluating more than one function at the same point will not necessarily be preserved in the additive limit; for instance the normalised ${\ell^2}$ norm

$\displaystyle (\frac{1}{|A_n|} \sum_{x \in A_n^m} |f_n(x)|^2)^{1/2}$

does not necessarily converge to the ${L^2}$ norm

$\displaystyle (\int_G |f(x)|^2\ d\mu_G(x))^{1/2},$

but can converge instead to a larger quantity, due to the presence of the orthogonal projection ${\pi_*}$ in the definition (4) of ${f}$.

An important special case of an additive limit occurs when the functions ${f_n\colon A_n^m \rightarrow {\bf C}}$ involved are indicator functions ${f_n = 1_{E_n}}$ of some subsets ${E_n}$ of ${A_n^m}$. The additive limit ${f \in L}$ does not necessarily remain an indicator function, but instead takes values in ${[0,1]}$ (much as a graphon ${p}$ takes values in ${[0,1]}$ even though the original indicators ${1_{E_n}}$ take values in ${\{0,1\}}$). The convolution ${f \star f\colon G \rightarrow [0,1]}$ is then the ultralimit of the normalised convolutions ${\frac{1}{|A_n|} 1_{E_n} \star 1_{E_n}}$; in particular, the measure of the support of ${f \star f}$ provides a lower bound on the limiting normalised cardinality ${\frac{1}{|A_n|} |E_n + E_n|}$ of a sumset. In many situations this lower bound is an equality, but this is not necessarily the case, because the sumset ${2E_n = E_n + E_n}$ could contain a large number of elements which have very few (${o(|A_n|)}$) representations as the sum of two elements of ${E_n}$, and in the limit these portions of the sumset fall outside of the support of ${f \star f}$. (One can think of the support of ${f \star f}$ as describing the “essential” sumset of ${2E_n = E_n + E_n}$, discarding those elements that have only very few representations.) Similarly for higher convolutions of ${f}$. Thus one can use additive limits to partially control the growth ${k E_n}$ of iterated sumsets of subsets ${E_n}$ of approximate groups ${A_n}$, in the regime where ${k}$ stays bounded and ${n}$ goes to infinity.

Theorem 1 can be proven by Fourier-analytic means (combined with Freiman’s theorem from additive combinatorics), and we will do so below the fold. For now, we give some illustrative examples of additive limits.

Example 2 (Bohr sets) We take ${A_n}$ to be the intervals ${A_n := \{ x \in {\bf Z}: |x| \leq N_n \}}$, where ${N_n}$ is a sequence going to infinity; these are ${2}$-approximate groups for all ${n}$. Let ${\theta}$ be an irrational real number, let ${I}$ be an interval in ${{\bf R}/{\bf Z}}$, and for each natural number ${n}$ let ${B_n}$ be the Bohr set

$\displaystyle B_n := \{ x \in A^{(n)}: \theta x \hbox{ mod } 1 \in I \}.$

In this case, the (reduced) Kronecker factor ${G}$ can be taken to be the infinite cylinder ${{\bf R} \times {\bf R}/{\bf Z}}$ with the usual Lebesgue measure ${\mu_G}$. The additive limits of ${1_{A_n}}$ and ${1_{B_n}}$ end up being ${1_A}$ and ${1_B}$, where ${A}$ is the finite cylinder

$\displaystyle A := \{ (x,t) \in {\bf R} \times {\bf R}/{\bf Z}: x \in [-1,1]\}$

and ${B}$ is the rectangle

$\displaystyle B := \{ (x,t) \in {\bf R} \times {\bf R}/{\bf Z}: x \in [-1,1]; t \in I \}.$

Geometrically, one should think of ${A_n}$ and ${B_n}$ as being wrapped around the cylinder ${{\bf R} \times {\bf R}/{\bf Z}}$ via the homomorphism ${x \mapsto (\frac{x}{N_n}, \theta x \hbox{ mod } 1)}$, and then one sees that ${B_n}$ is converging in some normalised weak sense to ${B}$, and similarly for ${A_n}$ and ${A}$. In particular, the additive limit predicts the growth rate of the iterated sumsets ${kB_n}$ to be quadratic in ${k}$ until ${k|I|}$ becomes comparable to ${1}$, at which point the growth transitions to linear growth, in the regime where ${k}$ is bounded and ${n}$ is large.

If ${\theta = \frac{p}{q}}$ were rational instead of irrational, then one would need to replace ${{\bf R}/{\bf Z}}$ by the finite subgroup ${\frac{1}{q}{\bf Z}/{\bf Z}}$ here.

Example 3 (Structured subsets of progressions) We take ${A_n}$ be the rank two progression

$\displaystyle A_n := \{ a + b N_n^2: a,b \in {\bf Z}; |a|, |b| \leq N_n \},$

where ${N_n}$ is a sequence going to infinity; these are ${4}$-approximate groups for all ${n}$. Let ${B_n}$ be the subset

$\displaystyle B_n := \{ a + b N_n^2: a,b \in {\bf Z}; |a|^2 + |b|^2 \leq N_n^2 \}.$

Then the (reduced) Kronecker factor can be taken to be ${G = {\bf R}^2}$ with Lebesgue measure ${\mu_G}$, and the additive limits of the ${1_{A_n}}$ and ${1_{B_n}}$ are then ${1_A}$ and ${1_B}$, where ${A}$ is the square

$\displaystyle A := \{ (a,b) \in {\bf R}^2: |a|, |b| \leq 1 \}$

and ${B}$ is the circle

$\displaystyle B := \{ (a,b) \in {\bf R}^2: a^2+b^2 \leq 1 \}.$

Geometrically, the picture is similar to the Bohr set one, except now one uses a Freiman homomorphism ${a + b N_n^2 \mapsto (\frac{a}{N_n}, \frac{b}{N_n})}$ for ${a,b = O( N_n )}$ to embed the original sets ${A_n, B_n}$ into the plane ${{\bf R}^2}$. In particular, one now expects the growth rate of the iterated sumsets ${k A_n}$ and ${k B_n}$ to be quadratic in ${k}$, in the regime where ${k}$ is bounded and ${n}$ is large.

Example 4 (Dissociated sets) Let ${d}$ be a fixed natural number, and take

$\displaystyle A_n = \{0, v_1,\dots,v_d,-v_1,\dots,-v_d \}$

where ${v_1,\dots,v_d}$ are randomly chosen elements of a large cyclic group ${{\bf Z}/p_n{\bf Z}}$, where ${p_n}$ is a sequence of primes going to infinity. These are ${O(d)}$-approximate groups. The (reduced) Kronecker factor ${G}$ can (almost surely) then be taken to be ${{\bf Z}^d}$ with counting measure, and the additive limit of ${1_{A_n}}$ is ${1_A}$, where ${A = \{ 0, e_1,\dots,e_d,-e_1,\dots,-e_d\}}$ and ${e_1,\dots,e_d}$ is the standard basis of ${{\bf Z}^d}$. In particular, the growth rates of ${k A_n}$ should grow approximately like ${k^d}$ for ${k}$ bounded and ${n}$ large.

Example 5 (Random subsets of groups) Let ${A_n = G_n}$ be a sequence of finite additive groups whose order is going to infinity. Let ${B_n}$ be a random subset of ${G_n}$ of some fixed density ${0 \leq \lambda \leq 1}$. Then (almost surely) the Kronecker factor here can be reduced all the way to the trivial group ${\{0\}}$, and the additive limit of the ${1_{B_n}}$ is the constant function ${\lambda}$. The convolutions ${\frac{1}{|G_n|} 1_{B_n} * 1_{B_n}}$ then converge in the ultralimit (modulo almost everywhere equivalence) to the pullback of ${\lambda^2}$; this reflects the fact that ${(1-o(1))|G_n|}$ of the elements of ${G_n}$ can be represented as the sum of two elements of ${B_n}$ in ${(\lambda^2 + o(1)) |G_n|}$ ways. In particular, ${B_n+B_n}$ occupies a proportion ${1-o(1)}$ of ${G_n}$.

Example 6 (Trigonometric series) Take ${A_n = G_n = {\bf Z}/p_n {\bf C}}$ for a sequence ${p_n}$ of primes going to infinity, and for each ${n}$ let ${\xi_{n,1},\xi_{n,2},\dots}$ be an infinite sequence of frequencies chosen uniformly and independently from ${{\bf Z}/p_n{\bf Z}}$. Let ${f_n\colon {\bf Z}/p_n{\bf Z} \rightarrow {\bf C}}$ denote the random trigonometric series

$\displaystyle f_n(x) := \sum_{j=1}^\infty 2^{-j} e^{2\pi i \xi_{n,j} x / p_n }.$

Then (almost surely) we can take the reduced Kronecker factor ${G}$ to be the infinite torus ${({\bf R}/{\bf Z})^{\bf N}}$ (with the Haar probability measure ${\mu_G}$), and the additive limit of the ${f_n}$ then becomes the function ${f\colon ({\bf R}/{\bf Z})^{\bf N} \rightarrow {\bf R}}$ defined by the formula

$\displaystyle f( (x_j)_{j=1}^\infty ) := \sum_{j=1}^\infty e^{2\pi i x_j}.$

In fact, the pullback ${\pi^* f}$ is the ultralimit of the ${f_n}$. As such, for any standard exponent ${1 \leq q < \infty}$, the normalised ${l^q}$ norm

$\displaystyle (\frac{1}{p_n} \sum_{x \in {\bf Z}/p_n{\bf Z}} |f_n(x)|^q)^{1/q}$

can be seen to converge to the limit

$\displaystyle (\int_{({\bf R}/{\bf Z})^{\bf N}} |f(x)|^q\ d\mu_G(x))^{1/q}.$

The reader is invited to consider combinations of the above examples, e.g. random subsets of Bohr sets, to get a sense of the general case of Theorem 1.

It is likely that this theorem can be extended to the noncommutative setting, using the noncommutative Freiman theorem of Emmanuel Breuillard, Ben Green, and myself, but I have not attempted to do so here (see though this recent preprint of Anush Tserunyan for some related explorations); in a separate direction, there should be extensions that can control higher Gowers norms, in the spirit of the work of Szegedy.

Note: the arguments below will presume some familiarity with additive combinatorics and with nonstandard analysis, and will be a little sketchy in places.

One of the first basic theorems in group theory is Cayley’s theorem, which links abstract finite groups with concrete finite groups (otherwise known as permutation groups).

Theorem 1 (Cayley’s theorem) Let ${G}$ be a group of some finite order ${n}$. Then ${G}$ is isomorphic to a subgroup ${\tilde G}$ of the symmetric group ${S_n}$ on ${n}$ elements ${\{1,\dots,n\}}$. Furthermore, this subgroup is simply transitive: given two elements ${x,y}$ of ${\{1,\dots,n\}}$, there is precisely one element ${\sigma}$ of ${\tilde G}$ such that ${\sigma(x)=y}$.

One can therefore think of ${S_n}$ as a sort of “universal” group that contains (up to isomorphism) all the possible groups of order ${n}$.

Proof: The group ${G}$ acts on itself by multiplication on the left, thus each element ${g \in G}$ may be identified with a permutation ${\sigma_g: G \rightarrow G}$ on ${G}$ given by the map ${\sigma_g(h) := gh}$. This can be easily verified to identify ${G}$ with a simply transitive permutation group on ${G}$. The claim then follows by arbitrarily identifying ${G}$ with ${\{1,\dots,n\}}$. $\Box$

More explicitly, the permutation group ${\tilde G}$ arises by arbitrarily enumerating ${G}$ as ${\{s_1,\dots,s_n\}}$ and then associating to each group element ${g \in G}$ the permutation ${\sigma_g: \{1,\dots,n\} \rightarrow \{1,\dots,n\}}$ defined by the formula

$\displaystyle g s_i = s_{\sigma_g(i)}.$

The simply transitive group ${\tilde G}$ given by Cayley’s theorem is not unique, due to the arbitrary choice of identification of ${G}$ with ${\{1,\dots,n\}}$, but is unique up to conjugation by an element of ${S_n}$. On the other hand, it is easy to see that every simply transitive subgroup of ${S_n}$ is of order ${n}$, and that two such groups are isomorphic if and only if they are conjugate by an element of ${S_n}$. Thus Cayley’s theorem in fact identifies the moduli space of groups of order ${n}$ (up to isomorphism) with the simply transitive subgroups of ${S_n}$ (up to conjugacy by elements of ${S_n}$).

One can generalise Cayley’s theorem to groups of infinite order without much difficulty. But in this post, I would like to note an (easy) generalisation of Cayley’s theorem in a different direction, in which the group ${G}$ is no longer assumed to be of order ${n}$, but rather to have an index ${n}$ subgroup that is isomorphic to a fixed group ${H}$. The generalisation is:

Theorem 2 (Cayley’s theorem for ${H}$-sets) Let ${H}$ be a group, and let ${G}$ be a group that contains an index ${n}$ subgroup isomorphic to ${H}$. Then ${G}$ is isomorphic to a subgroup ${\tilde G}$ of the semidirect product ${S_n \ltimes H^n}$, defined explicitly as the set of tuples ${(\sigma, (h_i)_{i=1}^n)}$ with product

$\displaystyle (\sigma, (h_i)_{i=1}^n) (\rho, (k_i)_{i=1}^n) := (\sigma \circ \rho, (h_{\rho(i)} k_i)_{i=1}^n )$

and inverse

$\displaystyle (\sigma, (h_i)_{i=1}^n)^{-1} := (\sigma^{-1}, (h_{\sigma(i)}^{-1})_{i=1}^n).$

(This group is a wreath product of ${H}$ with ${S_n}$, and is sometimes denoted ${H \wr S_n}$, or more precisely ${H \wr_{\{1,\dots,n\}} S_n}$.) Furthermore, ${\tilde G}$ is simply transitive in the following sense: given any two elements ${x,y}$ of ${\{1,\dots,n\}}$ and ${h,k \in H}$, there is precisely one ${(\sigma, (h_i)_{i=1}^n)}$ in ${\tilde G}$ such that ${\sigma(x)=y}$ and ${k = h_x h}$.

Of course, Theorem 1 is the special case of Theorem 2 when ${H}$ is trivial. This theorem allows one to view ${S_n \ltimes H^n}$ as a “universal” group for modeling all groups containing a copy of ${H}$ as an index ${n}$ subgroup, in exactly the same way that ${S_n}$ is a universal group for modeling groups of order ${n}$. This observation is not at all deep, but I had not seen it before, so I thought I would record it here. (EDIT: as pointed out in comments, this is a slight variant of the universal embedding theorem of Krasner and Kaloujnine, which covers the case when ${H}$ is normal, in which case one can embed ${G}$ into the wreath product ${H \wr G/H}$, which is a subgroup of ${H \wr S_n}$.)

Proof: The basic idea here is to replace the category of sets in Theorem 1 by the category of ${H}$-sets, by which we mean sets ${X}$ with a right-action of the group ${H}$. A morphism between two ${H}$-sets ${X,Y}$ is a function ${f: X \rightarrow Y}$ which respects the right action of ${H}$, thus ${f(x)h = f(xh)}$ for all ${x \in X}$ and ${h \in H}$.

Observe that if ${G}$ contains a copy of ${H}$ as a subgroup, then one can view ${G}$ as an ${H}$-set, using the right-action of ${H}$ (which we identify with the indicated subgroup of ${G}$). The left action of ${G}$ on itself commutes with the right-action of ${H}$, and so we can represent ${G}$ by ${H}$-set automorphisms on the ${H}$-set ${G}$.

As ${H}$ has index ${n}$ in ${G}$, we see that ${G}$ is (non-canonically) isomorphic (as an ${H}$-set) to the ${H}$-set ${\{1,\dots,n\} \times H}$ with the obvious right action of ${H}$: ${(i,h) k := (i,hk)}$. It is easy to see that the group of ${H}$-set automorphisms of ${\{1,\dots,n\} \times H}$ can be identified with ${S^n \ltimes H}$, with the latter group acting on the former ${H}$-set by the rule

$\displaystyle (\sigma, (h_i)_{i=1}^n) (i,h) := (\sigma(i), h_i h)$

(it is routine to verify that this is indeed an action of ${S^n \ltimes H}$ by ${H}$-set automorphisms. It is then a routine matter to verify the claims (the simple transitivity of ${\tilde G}$ follows from the simple transitivity of the action of ${G}$ on itself). $\Box$

More explicitly, the group ${\tilde G}$ arises by arbitrarily enumerating the left-cosets of ${H}$ in ${G}$ as ${\{s_1H,\dots,s_nH\}}$ and then associating to each group element ${g \in G}$ the element ${(\sigma_g, (h_{g,i})_{i=1}^n )}$, where the permutation ${\sigma_g: \{1,\dots,n\} \rightarrow \{1,\dots,n\}}$ and the elements ${h_{g,i} \in H}$ are defined by the formula

$\displaystyle g s_i = s_{\sigma_g(i)} h_{g,i}.$

By noting that ${H^n}$ is an index ${n!}$ normal subgroup of ${S_n \ltimes H^n}$, we recover the classical result of Poincaré that any group ${G}$ that contains ${H}$ as an index ${n}$ subgroup, contains a normal subgroup ${N}$ of index dividing ${n!}$ that is contained in ${H}$. (Quotienting out the ${H}$ right-action, we recover also the classical proof of this result, as the action of ${G}$ on itself then collapses to the action of ${G}$ on the quotient space ${G/H}$, the stabiliser of which is ${N}$.)

Exercise 1 Show that a simply transitive subgroup ${\tilde G}$ of ${S_n \ltimes H^n}$ contains a copy of ${H}$ as an index ${n}$ subgroup; in particular, there is a canonical embedding of ${H}$ into ${\tilde G}$, and ${\tilde G}$ can be viewed as an ${H}$-set.

Exercise 2 Show that any two simply transitive subgroups ${\tilde G_1, \tilde G_2}$ of ${S_n \ltimes H^n}$ are isomorphic simultaneously as groups and as ${H}$-sets (that is, there is a bijection ${\phi: \tilde G_1 \rightarrow \tilde G_2}$ that is simultaneously a group isomorphism and an ${H}$-set isomorphism) if and only if they are conjugate by an element of ${S_n \times H_n}$.

Analytic number theory is often concerned with the asymptotic behaviour of various arithmetic functions: functions ${f: {\bf N} \rightarrow {\bf R}}$ or ${f: {\bf N} \rightarrow {\bf C}}$ from the natural numbers ${{\bf N} = \{1,2,\dots\}}$ to the real numbers ${{\bf R}}$ or complex numbers ${{\bf C}}$. In this post, we will focus on the purely algebraic properties of these functions, and for reasons that will become clear later, it will be convenient to generalise the notion of an arithmetic function to functions ${f: {\bf N} \rightarrow R}$ taking values in some abstract commutative ring ${R}$. In this setting, we can add or multiply two arithmetic functions ${f,g: {\bf N} \rightarrow R}$ to obtain further arithmetic functions ${f+g, fg: {\bf N} \rightarrow R}$, and we can also form the Dirichlet convolution ${f*g: {\bf N} \rightarrow R}$ by the usual formula

$\displaystyle f*g(n) := \sum_{d|n} f(d) g(\frac{n}{d}).$

Regardless of what commutative ring ${R}$ is in used here, we observe that Dirichlet convolution is commutative, associative, and bilinear over ${R}$.

An important class of arithmetic functions in analytic number theory are the multiplicative functions, that is to say the arithmetic functions ${f: {\bf N} \rightarrow {\bf R}}$ such that ${f(1)=1}$ and

$\displaystyle f(nm) = f(n) f(m)$

for all coprime ${n,m \in {\bf N}}$. A subclass of these functions are the completely multiplicative functions, in which the restriction that ${n,m}$ be coprime is dropped. Basic examples of completely multiplicative functions (in the classical setting ${R={\bf C}}$) include

• the Kronecker delta ${\delta}$, defined by setting ${\delta(n)=1}$ for ${n=1}$ and ${\delta(n)=0}$ otherwise;
• the constant function ${1: n \mapsto 1}$ and the linear function ${n \mapsto n}$ (which by abuse of notation we denote by ${n}$);
• more generally monomials ${n \mapsto n^s}$ for any fixed complex number ${s}$ (in particular, the “Archimedean characters” ${n \mapsto n^{it}}$ for any fixed ${t \in {\bf R}}$), which by abuse of notation we denote by ${n^s}$;
• Dirichlet characters ${\chi}$;
• the Liouville function ${\lambda}$;
• the indicator function of the ${z}$smooth numbers (numbers whose prime factors are all at most ${z}$), for some given ${z}$; and
• the indicator function of the ${z}$rough numbers (numbers whose prime factors are all greater than ${z}$), for some given ${z}$.

Examples of multiplicative functions that are not completely multiplicative include

• the Möbius function ${\mu}$;
• the divisor function ${\tau}$ (also referred to as ${d}$);
• more generally, the higher order divisor functions ${\tau_k(n) = \sum_{d_1,\dots,d_k: d_1 \dots d_k = n} 1}$ for ${k \geq 1}$;
• the Euler totient function ${\phi}$;
• the number of roots ${n \mapsto \# \{ a \in {\bf Z}/n{\bf Z}: P(a) = 0\}}$ of a given polynomial ${P}$ defined over ${{\bf Z}}$;
• more generally, the point counting function ${n \mapsto V[{\bf Z}/n{\bf Z}]}$ of a given algebraic variety ${V}$ defined over ${{\bf Z}}$ (closely tied to the Hasse-Weil zeta function of ${V}$);
• the function ${r: n \mapsto r(n)}$ that counts the number of representations of ${n}$ as the sum of two squares;
• more generally, the function that maps a natural number ${n}$ to the number of ideals in a given number field ${K}$ of absolute norm ${n}$ (closely tied to the Dedekind zeta function of ${K}$).

These multiplicative functions interact well with the multiplication and convolution operations: if ${f,g: {\bf N} \rightarrow R}$ are multiplicative, then so are ${fg}$ and ${f * g}$, and if ${\psi}$ is completely multiplicative, then we also have

$\displaystyle \psi (f*g) = (\psi f) * (\psi g). \ \ \ \ \ (1)$

Finally, the product of completely multiplicative functions is again completely multiplicative. On the other hand, the sum of two multiplicative functions will never be multiplicative (just look at what happens at ${n=1}$), and the convolution of two completely multiplicative functions will usually just be multiplicative rather than completley multiplicative.

The specific multiplicative functions listed above are also related to each other by various important identities, for instance

$\displaystyle \delta * f = f; \quad \mu * 1 = \delta; \quad 1 * 1 = \tau; \quad \phi * 1 = n$

where ${f}$ is an arbitrary arithmetic function.

On the other hand, analytic number theory also is very interested in certain arithmetic functions that are not exactly multiplicative (and certainly not completely multiplicative). One particularly important such function is the von Mangoldt function ${\Lambda}$. This function is certainly not multiplicative, but is clearly closely related to such functions via such identities as ${\Lambda = \mu * L}$ and ${L = \Lambda * 1}$, where ${L: n\mapsto \log n}$ is the natural logarithm function. The purpose of this post is to point out that functions such as the von Mangoldt function lie in a class closely related to multiplicative functions, which I will call the derived multiplicative functions. More precisely:

Definition 1 A derived multiplicative function ${f: {\bf N} \rightarrow R}$ is an arithmetic function that can be expressed as the formal derivative

$\displaystyle f(n) = \frac{d}{d\varepsilon} F_\varepsilon(n) |_{\varepsilon=0}$

at the origin of a family ${f_\varepsilon: {\bf N}\rightarrow R}$ of multiplicative functions ${F_\varepsilon: {\bf N} \rightarrow R}$ parameterised by a formal parameter ${\varepsilon}$. Equivalently, ${f: {\bf N} \rightarrow R}$ is a derived multiplicative function if it is the ${\varepsilon}$ coefficient of a multiplicative function in the extension ${R[\varepsilon]/(\varepsilon^2)}$ of ${R}$ by a nilpotent infinitesimal ${\varepsilon}$; in other words, there exists an arithmetic function ${F: {\bf N} \rightarrow R}$ such that the arithmetic function ${F + \varepsilon f: {\bf N} \rightarrow R[\varepsilon]/(\varepsilon^2)}$ is multiplicative, or equivalently that ${F}$ is multiplicative and one has the Leibniz rule

$\displaystyle f(nm) = f(n) F(m) + F(n) f(m) \ \ \ \ \ (2)$

for all coprime ${n,m \in {\bf N}}$.

More generally, for any ${k\geq 0}$, a ${k}$-derived multiplicative function ${f: {\bf N} \rightarrow R}$ is an arithmetic function that can be expressed as the formal derivative

$\displaystyle f(n) = \frac{d^k}{d\varepsilon_1 \dots d\varepsilon_k} F_{\varepsilon_1,\dots,\varepsilon_k}(n) |_{\varepsilon_1,\dots,\varepsilon_k=0}$

at the origin of a family ${f_{\varepsilon_1,\dots,\varepsilon_k}: {\bf N} \rightarrow R}$ of multiplicative functions ${F_{\varepsilon_1,\dots,\varepsilon_k}: {\bf N} \rightarrow R}$ parameterised by formal parameters ${\varepsilon_1,\dots,\varepsilon_k}$. Equivalently, ${f}$ is the ${\varepsilon_1 \dots \varepsilon_k}$ coefficient of a multiplicative function in the extension ${R[\varepsilon_1,\dots,\varepsilon_k]/(\varepsilon_1^2,\dots,\varepsilon_k^2)}$ of ${R}$ by ${k}$ nilpotent infinitesimals ${\varepsilon_1,\dots,\varepsilon_k}$.

We define the notion of a ${k}$-derived completely multiplicative function similarly by replacing “multiplicative” with “completely multiplicative” in the above discussion.

There are Leibniz rules similar to (2) but they are harder to state; for instance, a doubly derived multiplicative function ${f: {\bf N} \rightarrow R}$ comes with singly derived multiplicative functions ${F_1, F_2: {\bf N} \rightarrow R}$ and a multiplicative function ${G: {\bf N} \rightarrow R}$ such that

$\displaystyle f(nm) = f(n) G(m) + F_1(n) F_2(m) + F_2(n) F_1(m) + G(n) f(m)$

for all coprime ${n,m \in {\bf N}}$.

One can then check that the von Mangoldt function ${\Lambda}$ is a derived multiplicative function, because ${\delta + \varepsilon \Lambda}$ is multiplicative in the ring ${{\bf C}[\varepsilon]/(\varepsilon^2)}$ with one infinitesimal ${\varepsilon}$. Similarly, the logarithm function ${L}$ is derived completely multiplicative because ${\exp( \varepsilon L ) := 1 + \varepsilon L}$ is completely multiplicative in ${{\bf C}[\varepsilon]/(\varepsilon^2)}$. More generally, any additive function ${\omega: {\bf N} \rightarrow R}$ is derived multiplicative because it is the top order coefficient of ${\exp(\varepsilon \omega) := 1 + \varepsilon \omega}$.

Remark 1 One can also phrase these concepts in terms of the formal Dirichlet series ${F(s) = \sum_n \frac{f(n)}{n^s}}$ associated to an arithmetic function ${f}$. A function ${f}$ is multiplicative if ${F}$ admits a (formal) Euler product; ${f}$ is derived multiplicative if ${F}$ is the (formal) first derivative of an Euler product with respect to some parameter (not necessarily ${s}$, although this is certainly an option); and so forth.

Using the definition of a ${k}$-derived multiplicative function as the top order coefficient of a multiplicative function of a ring with ${k}$ infinitesimals, it is easy to see that the product or convolution of a ${k}$-derived multiplicative function ${f: {\bf N} \rightarrow R}$ and a ${l}$-derived multiplicative function ${g: {\bf N} \rightarrow R}$ is necessarily a ${k+l}$-derived multiplicative function (again taking values in ${R}$). Thus, for instance, the higher-order von Mangoldt functions ${\Lambda_k := \mu * L^k}$ are ${k}$-derived multiplicative functions, because ${L^k}$ is a ${k}$-derived completely multiplicative function. More explicitly, ${L^k}$ is the top order coeffiicent of the completely multiplicative function ${\prod_{i=1}^k \exp(\varepsilon_i L)}$, and ${\Lambda_k}$ is the top order coefficient of the multiplicative function ${\mu * \prod_{i=1}^k \exp(\varepsilon_i L)}$, with both functions taking values in the ring ${C[\varepsilon_1,\dots,\varepsilon_k]/(\varepsilon_1^2,\dots,\varepsilon_k^2)}$ of complex numbers with ${k}$ infinitesimals ${\varepsilon_1,\dots,\varepsilon_k}$ attached.

It then turns out that most (if not all) of the basic identities used by analytic number theorists concerning derived multiplicative functions, can in fact be viewed as coefficients of identities involving purely multiplicative functions, with the latter identities being provable primarily from multiplicative identities, such as (1). This phenomenon is analogous to the one in linear algebra discussed in this previous blog post, in which many of the trace identities used there are derivatives of determinant identities. For instance, the Leibniz rule

$\displaystyle L (f * g) = (Lf)*g + f*(Lg)$

for any arithmetic functions ${f,g}$ can be viewed as the top order term in

$\displaystyle \exp(\varepsilon L) (f*g) = (\exp(\varepsilon L) f) * (\exp(\varepsilon L) g)$

in the ring with one infinitesimal ${\varepsilon}$, and then we see that the Leibniz rule is a special case (or a derivative) of (1), since ${\exp(\varepsilon L)}$ is completely multiplicative. Similarly, the formulae

$\displaystyle \Lambda = \mu * L; \quad L = \Lambda * 1$

are top order terms of

$\displaystyle (\delta + \varepsilon \Lambda) = \mu * \exp(\varepsilon L); \quad \exp(\varepsilon L) = (\delta + \varepsilon \Lambda) * 1,$

and the variant formula ${\Lambda = - (L\mu) * 1}$ is the top order term of

$\displaystyle (\delta + \varepsilon \Lambda) = (\exp(-\varepsilon L)\mu) * 1,$

which can then be deduced from the previous identities by noting that the completely multiplicative function ${\exp(-\varepsilon L)}$ inverts ${\exp(\varepsilon L)}$ multiplicatively, and also noting that ${L}$ annihilates ${\mu*1=\delta}$. The Selberg symmetry formula

$\displaystyle \Lambda_2 = \Lambda*\Lambda + \Lambda L, \ \ \ \ \ (3)$

which plays a key role in the Erdös-Selberg elementary proof of the prime number theorem (as discussed in this previous blog post), is the top order term of the identity

$\displaystyle \delta + \varepsilon_1 \Lambda + \varepsilon_2 \Lambda + \varepsilon_1\varepsilon_2 \Lambda_2 = (\exp(\varepsilon_2 L) (\delta + \varepsilon_1 \Lambda)) * (\delta + \varepsilon_2 \Lambda)$

involving the multiplicative functions ${\delta + \varepsilon_1 \Lambda + \varepsilon_2 \Lambda + \varepsilon_1\varepsilon_2 \Lambda_2}$, ${\exp(\varepsilon_2 L)}$, ${\delta+\varepsilon_1 \Lambda}$, ${\delta+\varepsilon_2 \Lambda}$ with two infinitesimals ${\varepsilon_1,\varepsilon_2}$, and this identity can be proven while staying purely within the realm of multiplicative functions, by using the identities

$\displaystyle \delta + \varepsilon_1 \Lambda + \varepsilon_2 \Lambda + \varepsilon_1\varepsilon_2 \Lambda_2 = \mu * (\exp(\varepsilon_1 L) \exp(\varepsilon_2 L))$

$\displaystyle \exp(\varepsilon_1 L) = 1 * (\delta + \varepsilon_1 \Lambda)$

$\displaystyle \delta + \varepsilon_2 \Lambda = \mu * \exp(\varepsilon_2 L)$

and (1). Similarly for higher identities such as

$\displaystyle \Lambda_3 = \Lambda L^2 + 3 \Lambda L * \Lambda + \Lambda * \Lambda * \Lambda$

which arise from expanding out ${\mu * (\exp(\varepsilon_1 L) \exp(\varepsilon_2 L) \exp(\varepsilon_3 L))}$ using (1) and the above identities; we leave this as an exercise to the interested reader.

An analogous phenomenon arises for identities that are not purely multiplicative in nature due to the presence of truncations, such as the Vaughan identity

$\displaystyle \Lambda_{> V} = \mu_{\leq U} * L - \mu_{\leq U} * \Lambda_{\leq V} * 1 + \mu_{>U} * \Lambda_{>V} * 1 \ \ \ \ \ (4)$

for any ${U,V \geq 1}$, where ${f_{>V} = f 1_{>V}}$ is the restriction of a multiplicative function ${f}$ to the natural numbers greater than ${V}$, and similarly for ${f_{\leq V}}$, ${f_{>U}}$, ${f_{\leq U}}$. In this particular case, (4) is the top order coefficient of the identity

$\displaystyle (\delta + \varepsilon \Lambda)_{>V} = \mu_{\leq U} * \exp(\varepsilon L) - \mu_{\leq U} * (\delta + \varepsilon \Lambda)_{\leq V} * 1$

$\displaystyle + \mu_{>U} * (\delta+\varepsilon \Lambda)_{>V} * 1$

which can be easily derived from the identities ${\delta = \mu_{\leq U} * 1 + \mu_{>U} * 1}$ and ${\exp(\varepsilon L) = (\delta + \varepsilon \Lambda)_{>V} * 1 + (\delta + \varepsilon \Lambda)_{\leq V} + 1}$. Similarly for the Heath-Brown identity

$\displaystyle \Lambda = \sum_{j=1}^K (-1)^{j-1} \binom{K}{j} \mu_{\leq U}^{*j} * 1^{*j-1} * L \ \ \ \ \ (5)$

valid for natural numbers up to ${U^K}$, where ${U \geq 1}$ and ${K \geq 1}$ are arbitrary parameters and ${f^{*j}}$ denotes the ${j}$-fold convolution of ${f}$, and discussed in this previous blog post; this is the top order coefficient of

$\displaystyle \delta + \varepsilon \Lambda = \sum_{j=1}^K (-1)^{j-1} \binom{K}{j} \mu_{\leq U}^{*j} * 1^{*j-1} * \exp( \varepsilon L )$

and arises by first observing that

$\displaystyle (\mu - \mu_{\leq U})^{*K} * 1^{*K-1} * \exp(\varepsilon L) = \mu_{>U}^{*K} * 1^{*K-1} * \exp( \varepsilon L )$

vanishes up to ${U^K}$, and then expanding the left-hand side using the binomial formula and the identity ${\mu^{*K} * 1^{*K-1} * \exp(\varepsilon L) = \delta + \varepsilon \Lambda}$.

One consequence of this phenomenon is that identities involving derived multiplicative functions tend to have a dimensional consistency property: all terms in the identity have the same order of derivation in them. For instance, all the terms in the Selberg symmetry formula (3) are doubly derived functions, all the terms in the Vaughan identity (4) or the Heath-Brown identity (5) are singly derived functions, and so forth. One can then use dimensional analysis to help ensure that one has written down a key identity involving such functions correctly, much as is done in physics.

In addition to the dimensional analysis arising from the order of derivation, there is another dimensional analysis coming from the value of multiplicative functions at primes ${p}$ (which is more or less equivalent to the order of pole of the Dirichlet series at ${s=1}$). Let us say that a multiplicative function ${f: {\bf N} \rightarrow R}$ has a pole of order ${j}$ if one has ${f(p)=j}$ on the average for primes ${p}$, where we will be a bit vague as to what “on the average” means as it usually does not matter in applications. Thus for instance, ${1}$ or ${\exp(\varepsilon L)}$ has a pole of order ${1}$ (a simple pole), ${\delta}$ or ${\delta + \varepsilon \Lambda}$ has a pole of order ${0}$ (i.e. neither a zero or a pole), Dirichlet characters also have a pole of order ${0}$ (although this is slightly nontrivial, requiring Dirichlet’s theorem), ${\mu}$ has a pole of order ${-1}$ (a simple zero), ${\tau}$ has a pole of order ${2}$, and so forth. Note that the convolution of a multiplicative function with a pole of order ${j}$ with a multiplicative function with a pole of order ${j'}$ will be a multiplicative function with a pole of order ${j+j'}$. If there is no oscillation in the primes ${p}$ (e.g. if ${f(p)=j}$ for all primes ${p}$, rather than on the average), it is also true that the product of a multiplicative function with a pole of order ${j}$ with a multiplicative function with a pole of order ${j'}$ will be a multiplicative function with a pole of order ${jj'}$. The situation is significantly different though in the presence of oscillation; for instance, if ${\chi}$ is a quadratic character then ${\chi^2}$ has a pole of order ${1}$ even though ${\chi}$ has a pole of order ${0}$.

A ${k}$-derived multiplicative function will then be said to have an underived pole of order ${j}$ if it is the top order coefficient of a multiplicative function with a pole of order ${j}$; in terms of Dirichlet series, this roughly means that the Dirichlet series has a pole of order ${j+k}$ at ${s=1}$. For instance, the singly derived multiplicative function ${\Lambda}$ has an underived pole of order ${0}$, because it is the top order coefficient of ${\delta + \varepsilon \Lambda}$, which has a pole of order ${0}$; similarly ${L}$ has an underived pole of order ${1}$, being the top order coefficient of ${\exp(\varepsilon L)}$. More generally, ${\Lambda_k}$ and ${L^k}$ have underived poles of order ${0}$ and ${1}$ respectively for any ${k}$.

By taking top order coefficients, we then see that the convolution of a ${k}$-derived multiplicative function with underived pole of order ${j}$ and a ${k'}$-derived multiplicative function with underived pole of order ${j'}$ is a ${k+k'}$-derived multiplicative function with underived pole of order ${j+j'}$. If there is no oscillation in the primes, the product of these functions will similarly have an underived pole of order ${jj'}$, for instance ${\Lambda L}$ has an underived pole of order ${0}$. We then have the dimensional consistency property that in any of the standard identities involving derived multiplicative functions, all terms not only have the same derived order, but also the same underived pole order. For instance, in (3), (4), (5) all terms have underived pole order ${0}$ (with any Mobius function terms being counterbalanced by a matching term of ${1}$ or ${L}$). This gives a second way to use dimensional analysis as a consistency check. For instance, any identity that involves a linear combination of ${\mu_{\leq U} * L}$ and ${\Lambda_{>V} * 1}$ is suspect because the underived pole orders do not match (being ${0}$ and ${1}$ respectively), even though the derived orders match (both are ${1}$).

One caveat, though: this latter dimensional consistency breaks down for identities that involve infinitely many terms, such as Linnik’s identity

$\displaystyle \Lambda = \sum_{i=0}^\infty (-1)^{i} L * 1_{>1}^{*i}.$

In this case, one can still rewrite things in terms of multiplicative functions as

$\displaystyle \delta + \varepsilon \Lambda = \sum_{i=0}^\infty (-1)^i \exp(\varepsilon L) * 1_{>1}^{*i},$

so the former dimensional consistency is still maintained.

I thank Andrew Granville, Kannan Soundararajan, and Emmanuel Kowalski for helpful conversations on these topics.

In the traditional foundations of probability theory, one selects a probability space ${(\Omega, {\mathcal B}, {\mathbf P})}$, and makes a distinction between deterministic mathematical objects, which do not depend on the sampled state ${\omega \in \Omega}$, and stochastic (or random) mathematical objects, which do depend (but in a measurable fashion) on the sampled state ${\omega \in \Omega}$. For instance, a deterministic real number would just be an element ${x \in {\bf R}}$, whereas a stochastic real number (or real random variable) would be a measurable function ${x: \Omega \rightarrow {\bf R}}$, where in this post ${{\bf R}}$ will always be endowed with the Borel ${\sigma}$-algebra. (For readers familiar with nonstandard analysis, the adjectives “deterministic” and “stochastic” will be used here in a manner analogous to the uses of the adjectives “standard” and “nonstandard” in nonstandard analysis. The analogy is particularly close when comparing with the “cheap nonstandard analysis” discussed in this previous blog post. We will also use “relative to ${\Omega}$” as a synonym for “stochastic”.)

Actually, for our purposes we will adopt the philosophy of identifying stochastic objects that agree almost surely, so if one was to be completely precise, we should define a stochastic real number to be an equivalence class ${[x]}$ of measurable functions ${x: \Omega \rightarrow {\bf R}}$, up to almost sure equivalence. However, we shall often abuse notation and write ${[x]}$ simply as ${x}$.

More generally, given any measurable space ${X = (X, {\mathcal X})}$, we can talk either about deterministic elements ${x \in X}$, or about stochastic elements of ${X}$, that is to say equivalence classes ${[x]}$ of measurable maps ${x: \Omega \rightarrow X}$ up to almost sure equivalence. We will use ${\Gamma(X|\Omega)}$ to denote the set of all stochastic elements of ${X}$. (For readers familiar with sheaves, it may helpful for the purposes of this post to think of ${\Gamma(X|\Omega)}$ as the space of measurable global sections of the trivial ${X}$bundle over ${\Omega}$.) Of course every deterministic element ${x}$ of ${X}$ can also be viewed as a stochastic element ${x|\Omega \in \Gamma(X|\Omega)}$ given by (the equivalence class of) the constant function ${\omega \mapsto x}$, thus giving an embedding of ${X}$ into ${\Gamma(X|\Omega)}$. We do not attempt here to give an interpretation of ${\Gamma(X|\Omega)}$ for sets ${X}$ that are not equipped with a ${\sigma}$-algebra ${{\mathcal X}}$.

Remark 1 In my previous post on the foundations of probability theory, I emphasised the freedom to extend the sample space ${(\Omega, {\mathcal B}, {\mathbf P})}$ to a larger sample space whenever one wished to inject additional sources of randomness. This is of course an important freedom to possess (and in the current formalism, is the analogue of the important operation of base change in algebraic geometry), but in this post we will focus on a single fixed sample space ${(\Omega, {\mathcal B}, {\mathbf P})}$, and not consider extensions of this space, so that one only has to consider two types of mathematical objects (deterministic and stochastic), as opposed to having many more such types, one for each potential choice of sample space (with the deterministic objects corresponding to the case when the sample space collapses to a point).

Any (measurable) ${k}$-ary operation on deterministic mathematical objects then extends to their stochastic counterparts by applying the operation pointwise. For instance, the addition operation ${+: {\bf R} \times {\bf R} \rightarrow {\bf R}}$ on deterministic real numbers extends to an addition operation ${+: \Gamma({\bf R}|\Omega) \times \Gamma({\bf R}|\Omega) \rightarrow \Gamma({\bf R}|\Omega)}$, by defining the class ${[x]+[y]}$ for ${x,y: \Omega \rightarrow {\bf R}}$ to be the equivalence class of the function ${\omega \mapsto x(\omega) + y(\omega)}$; this operation is easily seen to be well-defined. More generally, any measurable ${k}$-ary deterministic operation ${O: X_1 \times \dots \times X_k \rightarrow Y}$ between measurable spaces ${X_1,\dots,X_k,Y}$ extends to an stochastic operation ${O: \Gamma(X_1|\Omega) \times \dots \Gamma(X_k|\Omega) \rightarrow \Gamma(Y|\Omega)}$ in the obvious manner.

There is a similar story for ${k}$-ary relations ${R: X_1 \times \dots \times X_k \rightarrow \{\hbox{true},\hbox{false}\}}$, although here one has to make a distinction between a deterministic reading of the relation and a stochastic one. Namely, if we are given stochastic objects ${x_i \in \Gamma(X_i|\Omega)}$ for ${i=1,\dots,k}$, the relation ${R(x_1,\dots,x_k)}$ does not necessarily take values in the deterministic Boolean algebra ${\{ \hbox{true}, \hbox{false}\}}$, but only in the stochastic Boolean algebra ${\Gamma(\{ \hbox{true}, \hbox{false}\}|\Omega)}$ – thus ${R(x_1,\dots,x_k)}$ may be true with some positive probability and also false with some positive probability (with the event that ${R(x_1,\dots,x_k)}$ being stochastically true being determined up to null events). Of course, the deterministic Boolean algebra embeds in the stochastic one, so we can talk about a relation ${R(x_1,\dots,x_k)}$ being determinstically true or deterministically false, which (due to our identification of stochastic objects that agree almost surely) means that ${R(x_1(\omega),\dots,x_k(\omega))}$ is almost surely true or almost surely false respectively. For instance given two stochastic objects ${x,y}$, one can view their equality relation ${x=y}$ as having a stochastic truth value. This is distinct from the way the equality symbol ${=}$ is used in mathematical logic, which we will now call “equality in the deterministic sense” to reduce confusion. Thus, ${x=y}$ in the deterministic sense if and only if the stochastic truth value of ${x=y}$ is equal to ${\hbox{true}}$, that is to say that ${x(\omega)=y(\omega)}$ for almost all ${\omega}$.

Any universal identity for deterministic operations (or universal implication between identities) extends to their stochastic counterparts: for instance, addition is commutative, associative, and cancellative on the space of deterministic reals ${{\bf R}}$, and is therefore commutative, associative, and cancellative on stochastic reals ${\Gamma({\bf R}|\Omega)}$ as well. However, one has to be more careful when working with mathematical laws that are not expressible as universal identities, or implications between identities. For instance, ${{\bf R}}$ is an integral domain: if ${x_1,x_2 \in {\bf R}}$ are deterministic reals such that ${x_1 x_2=0}$, then one must have ${x_1=0}$ or ${x_2=0}$. However, if ${x_1, x_2 \in \Gamma({\bf R}|\Omega)}$ are stochastic reals such that ${x_1 x_2 = 0}$ (in the deterministic sense), then it is no longer necessarily the case that ${x_1=0}$ (in the deterministic sense) or that ${x_2=0}$ (in the deterministic sense); however, it is still true that “${x_1=0}$ or ${x_2=0}$” is true in the deterministic sense if one interprets the boolean operator “or” stochastically, thus “${x_1(\omega)=0}$ or ${x_2(\omega)=0}$” is true for almost all ${\omega}$. Another way to properly obtain a stochastic interpretation of the integral domain property of ${{\bf R}}$ is to rewrite it as

$\displaystyle x_1,x_2 \in {\bf R}, x_1 x_2 = 0 \implies x_i=0 \hbox{ for some } i \in \{1,2\}$

and then make all sets stochastic to obtain the true statement

$\displaystyle x_1,x_2 \in \Gamma({\bf R}|\Omega), x_1 x_2 = 0 \implies x_i=0 \hbox{ for some } i \in \Gamma(\{1,2\}|\Omega),$

thus we have to allow the index ${i}$ for which vanishing ${x_i=0}$ occurs to also be stochastic, rather than deterministic. (A technical note: when one proves this statement, one has to select ${i}$ in a measurable fashion; for instance, one can choose ${i(\omega)}$ to equal ${1}$ when ${x_1(\omega)=0}$, and ${2}$ otherwise (so that in the “tie-breaking” case when ${x_1(\omega)}$ and ${x_2(\omega)}$ both vanish, one always selects ${i(\omega)}$ to equal ${1}$).)

Similarly, the law of the excluded middle fails when interpreted deterministically, but remains true when interpreted stochastically: if ${S}$ is a stochastic statement, then it is not necessarily the case that ${S}$ is either deterministically true or deterministically false; however the sentence “${S}$ or not-${S}$” is still deterministically true if the boolean operator “or” is interpreted stochastically rather than deterministically.

To avoid having to keep pointing out which operations are interpreted stochastically and which ones are interpreted deterministically, we will use the following convention: if we assert that a mathematical sentence ${S}$ involving stochastic objects is true, then (unless otherwise specified) we mean that ${S}$ is deterministically true, assuming that all relations used inside ${S}$ are interpreted stochastically. For instance, if ${x,y}$ are stochastic reals, when we assert that “Exactly one of ${x < y}$, ${x=y}$, or ${x>y}$ is true”, then by default it is understood that the relations ${<}$, ${=}$, ${>}$ and the boolean operator “exactly one of” are interpreted stochastically, and the assertion is that the sentence is deterministically true.

In the above discussion, the stochastic objects ${x}$ being considered were elements of a deterministic space ${X}$, such as the reals ${{\bf R}}$. However, it can often be convenient to generalise this situation by allowing the ambient space ${X}$ to also be stochastic. For instance, one might wish to consider a stochastic vector ${v(\omega)}$ inside a stochastic vector space ${V(\omega)}$, or a stochastic edge ${e}$ of a stochastic graph ${G(\omega)}$. In order to formally describe this situation within the classical framework of measure theory, one needs to place all the ambient spaces ${X(\omega)}$ inside a measurable space. This can certainly be done in many contexts (e.g. when considering random graphs on a deterministic set of vertices, or if one is willing to work up to equivalence and place the ambient spaces inside a suitable moduli space), but is not completely natural in other contexts. For instance, if one wishes to consider stochastic vector spaces of potentially unbounded dimension (in particular, potentially larger than any given cardinal that one might specify in advance), then the class of all possible vector spaces is so large that it becomes a proper class rather than a set (even if one works up to equivalence), making it problematic to give this class the structure of a measurable space; furthermore, even once one does so, one needs to take additional care to pin down what it would mean for a random vector ${\omega \mapsto v_\omega}$ lying in a random vector space ${\omega \mapsto V_\omega}$ to depend “measurably” on ${\omega}$.

Of course, in any reasonable application one can avoid the set theoretic issues at least by various ad hoc means, for instance by restricting the dimension of all spaces involved to some fixed cardinal such as ${2^{\aleph_0}}$. However, the measure-theoretic issues can require some additional effort to resolve properly.

In this post I would like to describe a different way to formalise stochastic spaces, and stochastic elements of these spaces, by viewing the spaces as measure-theoretic analogue of a sheaf, but being over the probability space ${\Omega}$ rather than over a topological space; stochastic objects are then sections of such sheaves. Actually, for minor technical reasons it is convenient to work in the slightly more general setting in which the base space ${\Omega}$ is a finite measure space ${(\Omega, {\mathcal B}, \mu)}$ rather than a probability space, thus ${\mu(\Omega)}$ can take any value in ${[0,+\infty)}$ rather than being normalised to equal ${1}$. This will allow us to easily localise to subevents ${\Omega'}$ of ${\Omega}$ without the need for normalisation, even when ${\Omega'}$ is a null event (though we caution that the map ${x \mapsto x|\Omega'}$ from deterministic objects ${x}$ ceases to be injective in this latter case). We will however still continue to use probabilistic terminology. despite the lack of normalisation; thus for instance, sets ${E}$ in ${{\mathcal B}}$ will be referred to as events, the measure ${\mu(E)}$ of such a set will be referred to as the probability (which is now permitted to exceed ${1}$ in some cases), and an event whose complement is a null event shall be said to hold almost surely. It is in fact likely that almost all of the theory below extends to base spaces which are ${\sigma}$-finite rather than finite (for instance, by damping the measure to become finite, without introducing any further null events), although we will not pursue this further generalisation here.

The approach taken in this post is “topos-theoretic” in nature (although we will not use the language of topoi explicitly here), and is well suited to a “pointless” or “point-free” approach to probability theory, in which the role of the stochastic state ${\omega \in \Omega}$ is suppressed as much as possible; instead, one strives to always adopt a “relative point of view”, with all objects under consideration being viewed as stochastic objects relative to the underlying base space ${\Omega}$. In this perspective, the stochastic version of a set is as follows.

Definition 1 (Stochastic set) Unless otherwise specified, we assume that we are given a fixed finite measure space ${\Omega = (\Omega, {\mathcal B}, \mu)}$ (which we refer to as the base space). A stochastic set (relative to ${\Omega}$) is a tuple ${X|\Omega = (\Gamma(X|E)_{E \in {\mathcal B}}, ((|E))_{E \subset F, E,F \in {\mathcal B}})}$ consisting of the following objects:

• A set ${\Gamma(X|E)}$ assigned to each event ${E \in {\mathcal B}}$; and
• A restriction map ${x \mapsto x|E}$ from ${\Gamma(X|F)}$ to ${\Gamma(X|E)}$ to each pair ${E \subset F}$ of nested events ${E,F \in {\mathcal B}}$. (Strictly speaking, one should indicate the dependence on ${F}$ in the notation for the restriction map, e.g. using ${x \mapsto x|(E \leftarrow F)}$ instead of ${x \mapsto x|E}$, but we will abuse notation by omitting the ${F}$ dependence.)

We refer to elements of ${\Gamma(X|E)}$ as local stochastic elements of the stochastic set ${X|\Omega}$, localised to the event ${E}$, and elements of ${\Gamma(X|\Omega)}$ as global stochastic elements (or simply elements) of the stochastic set. (In the language of sheaves, one would use “sections” instead of “elements” here, but I prefer to use the latter terminology here, for compatibility with conventional probabilistic notation, where for instance measurable maps from ${\Omega}$ to ${{\bf R}}$ are referred to as real random variables, rather than sections of the reals.)

Furthermore, we impose the following axioms:

• (Category) The map ${x \mapsto x|E}$ from ${\Gamma(X|E)}$ to ${\Gamma(X|E)}$ is the identity map, and if ${E \subset F \subset G}$ are events in ${{\mathcal B}}$, then ${((x|F)|E) = (x|E)}$ for all ${x \in \Gamma(X|G)}$.
• (Null events trivial) If ${E \in {\mathcal B}}$ is a null event, then the set ${\Gamma(X|E)}$ is a singleton set. (In particular, ${\Gamma(X|\emptyset)}$ is always a singleton set; this is analogous to the convention that ${x^0=1}$ for any number ${x}$.)
• (Countable gluing) Suppose that for each natural number ${n}$, one has an event ${E_n \in {\mathcal B}}$ and an element ${x_n \in \Gamma(X|E_n)}$ such that ${x_n|(E_n \cap E_m) = x_m|(E_n \cap E_m)}$ for all ${n,m}$. Then there exists a unique ${x\in \Gamma(X|\bigcup_{n=1}^\infty E_n)}$ such that ${x_n = x|E_n}$ for all ${n}$.

If ${\Omega'}$ is an event in ${\Omega}$, we define the localisation ${X|\Omega'}$ of the stochastic set ${X|\Omega}$ to ${\Omega'}$ to be the stochastic set

$\displaystyle X|\Omega' := (\Gamma(X|E)_{E \in {\mathcal B}; E \subset \Omega'}, ((|E))_{E \subset F \subset \Omega', E,F \in {\mathcal B}})$

relative to ${\Omega'}$. (Note that there is no need to renormalise the measure on ${\Omega'}$, as we are not demanding that our base space have total measure ${1}$.)

The following fact is useful for actually verifying that a given object indeed has the structure of a stochastic set:

Exercise 1 Show that to verify the countable gluing axiom of a stochastic set, it suffices to do so under the additional hypothesis that the events ${E_n}$ are disjoint. (Note that this is quite different from the situation with sheaves over a topological space, in which the analogous gluing axiom is often trivial in the disjoint case but has non-trivial content in the overlapping case. This is ultimately because a ${\sigma}$-algebra is closed under all Boolean operations, whereas a topology is only closed under union and intersection.)

Let us illustrate the concept of a stochastic set with some examples.

Example 1 (Discrete case) A simple case arises when ${\Omega}$ is a discrete space which is at most countable. If we assign a set ${X_\omega}$ to each ${\omega \in \Omega}$, with ${X_\omega}$ a singleton if ${\mu(\{\omega\})=0}$. One then sets ${\Gamma(X|E) := \prod_{\omega \in E} X_\omega}$, with the obvious restriction maps, giving rise to a stochastic set ${X|\Omega}$. (Thus, a local element ${x}$ of ${\Gamma(X|E)}$ can be viewed as a map ${\omega \mapsto x(\omega)}$ on ${E}$ that takes values in ${X_\omega}$ for each ${\omega \in E}$.) Conversely, it is not difficult to see that any stochastic set over an at most countable discrete probability space ${\Omega}$ is of this form up to isomorphism. In this case, one can think of ${X|\Omega}$ as a bundle of sets ${X_\omega}$ over each point ${\omega}$ (of positive probability) in the base space ${\Omega}$. One can extend this bundle interpretation of stochastic sets to reasonably nice sample spaces ${\Omega}$ (such as standard Borel spaces) and similarly reasonable ${X}$; however, I would like to avoid this interpretation in the formalism below in order to be able to easily work in settings in which ${\Omega}$ and ${X}$ are very “large” (e.g. not separable in any reasonable sense). Note that we permit some of the ${X_\omega}$ to be empty, thus it can be possible for ${\Gamma(X|\Omega)}$ to be empty whilst ${\Gamma(X|E)}$ for some strict subevents ${E}$ of ${\Omega}$ to be non-empty. (This is analogous to how it is possible for a sheaf to have local sections but no global sections.) As such, the space ${\Gamma(X|\Omega)}$ of global elements does not completely determine the stochastic set ${X|\Omega}$; one sometimes needs to localise to an event ${E}$ in order to see the full structure of such a set. Thus it is important to distinguish between a stochastic set ${X|\Omega}$ and its space ${\Gamma(X|\Omega)}$ of global elements. (As such, it is a slight abuse of the axiom of extensionality to refer to global elements of ${X|\Omega}$ simply as “elements”, but hopefully this should not cause too much confusion.)

Example 2 (Measurable spaces as stochastic sets) Returning now to a general base space ${\Omega}$, any (deterministic) measurable space ${X}$ gives rise to a stochastic set ${X|\Omega}$, with ${\Gamma(X|E)}$ being defined as in previous discussion as the measurable functions from ${E}$ to ${X}$ modulo almost everywhere equivalence (in particular, ${\Gamma(X|E)}$ a singleton set when ${E}$ is null), with the usual restriction maps. The constraint of measurability on the maps ${x: E \rightarrow \Omega}$, together with the quotienting by almost sure equivalence, means that ${\Gamma(X|E)}$ is now more complicated than a plain Cartesian product ${\prod_{\omega \in E} X_\omega}$ of fibres, but this still serves as a useful first approximation to what ${\Gamma(X|E)}$ is for the purposes of developing intuition. Indeed, the measurability constraint is so weak (as compared for instance to topological or smooth constraints in other contexts, such as sheaves of continuous or smooth sections of bundles) that the intuition of essentially independent fibres is quite an accurate one, at least if one avoids consideration of an uncountable number of objects simultaneously.

Example 3 (Extended Hilbert modules) This example is the one that motivated this post for me. Suppose that one has an extension ${(\tilde \Omega, \tilde {\mathcal B}, \tilde \mu)}$ of the base space ${(\Omega, {\mathcal B},\mu)}$, thus we have a measurable factor map ${\pi: \tilde \Omega \rightarrow \Omega}$ such that the pushforward of the measure ${\tilde \mu}$ by ${\pi}$ is equal to ${\mu}$. Then we have a conditional expectation operator ${\pi_*: L^2(\tilde \Omega,\tilde {\mathcal B},\tilde \mu) \rightarrow L^2(\Omega,{\mathcal B},\mu)}$, defined as the adjoint of the pullback map ${\pi^*: L^2(\Omega,{\mathcal B},\mu) \rightarrow L^2(\tilde \Omega,\tilde {\mathcal B},\tilde \mu)}$. As is well known, the conditional expectation operator also extends to a contraction ${\pi_*: L^1(\tilde \Omega,\tilde {\mathcal B},\tilde \mu) \rightarrow L^1(\Omega,{\mathcal B}, \mu)}$; by monotone convergence we may also extend ${\pi_*}$ to a map from measurable functions from ${\tilde \Omega}$ to the extended non-negative reals ${[0,+\infty]}$, to measurable functions from ${\Omega}$ to ${[0,+\infty]}$. We then define the “extended Hilbert module” ${L^2(\tilde \Omega|\Omega)}$ to be the space of functions ${f \in L^2(\tilde \Omega,\tilde {\mathcal B},\tilde \mu)}$ with ${\pi_*(|f|^2)}$ finite almost everywhere. This is an extended version of the Hilbert module ${L^\infty_{\Omega} L^2(\tilde \Omega|\Omega)}$, which is defined similarly except that ${\pi_*(|f|^2)}$ is required to lie in ${L^\infty(\Omega,{\mathcal B},\mu)}$; this is a Hilbert module over ${L^\infty(\Omega, {\mathcal B}, \mu)}$ which is of particular importance in the Furstenberg-Zimmer structure theory of measure-preserving systems. We can then define the stochastic set ${L^2_\pi(\tilde \Omega)|\Omega}$ by setting

$\displaystyle \Gamma(L^2_\pi(\tilde \Omega)|E) := L^2( \pi^{-1}(E) | E )$

with the obvious restriction maps. In the case that ${\Omega,\Omega'}$ are standard Borel spaces, one can disintegrate ${\mu'}$ as an integral ${\mu' = \int_\Omega \nu_\omega\ d\mu(\omega)}$ of probability measures ${\nu_\omega}$ (supported in the fibre ${\pi^{-1}(\{\omega\})}$), in which case this stochastic set can be viewed as having fibres ${L^2( \tilde \Omega, \tilde {\mathcal B}, \nu_\omega )}$ (though if ${\Omega}$ is not discrete, there are still some measurability conditions in ${\omega}$ on the local and global elements that need to be imposed). However, I am interested in the case when ${\Omega,\Omega'}$ are not standard Borel spaces (in fact, I will take them to be algebraic probability spaces, as defined in this previous post), in which case disintegrations are not available. However, it appears that the stochastic analysis developed in this blog post can serve as a substitute for the tool of disintegration in this context.

We make the remark that if ${X|\Omega}$ is a stochastic set and ${E, F}$ are events that are equivalent up to null events, then one can identify ${\Gamma(X|E)}$ with ${\Gamma(X|F)}$ (through their common restriction to ${\Gamma(X|(E \cap F))}$, with the restriction maps now being bijections). As such, the notion of a stochastic set does not require the full structure of a concrete probability space ${(\Omega, {\mathcal B}, {\mathbf P})}$; one could also have defined the notion using only the abstract ${\sigma}$-algebra consisting of ${{\mathcal B}}$ modulo null events as the base space, or equivalently one could define stochastic sets over the algebraic probability spaces defined in this previous post. However, we will stick with the classical formalism of concrete probability spaces here so as to keep the notation reasonably familiar.

As a corollary of the above observation, we see that if the base space ${\Omega}$ has total measure ${0}$, then all stochastic sets are trivial (they are just points).

Exercise 2 If ${X|\Omega}$ is a stochastic set, show that there exists an event ${\Omega'}$ with the property that for any event ${E}$, ${\Gamma(X|E)}$ is non-empty if and only if ${E}$ is contained in ${\Omega'}$ modulo null events. (In particular, ${\Omega'}$ is unique up to null events.) Hint: consider the numbers ${\mu( E )}$ for ${E}$ ranging over all events with ${\Gamma(X|E)}$ non-empty, and form a maximising sequence for these numbers. Then use all three axioms of a stochastic set.

One can now start take many of the fundamental objects, operations, and results in set theory (and, hence, in most other categories of mathematics) and establish analogues relative to a finite measure space. Implicitly, what we will be doing in the next few paragraphs is endowing the category of stochastic sets with the structure of an elementary topos. However, to keep things reasonably concrete, we will not explicitly emphasise the topos-theoretic formalism here, although it is certainly lurking in the background.

Firstly, we define a stochastic function ${f: X|\Omega \rightarrow Y|\Omega}$ between two stochastic sets ${X|\Omega, Y|\Omega}$ to be a collection of maps ${f: \Gamma(X|E) \rightarrow \Gamma(Y|E)}$ for each ${E \in {\mathcal B}}$ which form a natural transformation in the sense that ${f(x|E) = f(x)|E}$ for all ${x \in \Gamma(X|F)}$ and nested events ${E \subset F}$. In the case when ${\Omega}$ is discrete and at most countable (and after deleting all null points), a stochastic function is nothing more than a collection of functions ${f_\omega: X_\omega \rightarrow Y_\omega}$ for each ${\omega \in \Omega}$, with the function ${f: \Gamma(X|E) \rightarrow \Gamma(Y|E)}$ then being a direct sum of the factor functions ${f_\omega}$:

$\displaystyle f( (x_\omega)_{\omega \in E} ) = ( f_\omega(x_\omega) )_{\omega \in E}.$

Thus (in the discrete, at most countable setting, at least) stochastic functions do not mix together information from different states ${\omega}$ in a sample space; the value of ${f(x)}$ at ${\omega}$ depends only on the value of ${x}$ at ${\omega}$. The situation is a bit more subtle for continuous probability spaces, due to the identification of stochastic objects that agree almost surely, nevertheness it is still good intuition to think of stochastic functions as essentially being “pointwise” or “local” in nature.

One can now form the stochastic set ${\hbox{Hom}(X \rightarrow Y)|\Omega}$ of functions from ${X|\Omega}$ to ${Y|\Omega}$, by setting ${\Gamma(\hbox{Hom}(X \rightarrow Y)|E)}$ for any event ${E}$ to be the set of local stochastic functions ${f: X|E \rightarrow Y|E}$ of the localisations of ${X|\Omega, Y|\Omega}$ to ${E}$; this is a stochastic set if we use the obvious restriction maps. In the case when ${\Omega}$ is discrete and at most countable, the fibre ${\hbox{Hom}(X \rightarrow Y)_\omega}$ at a point ${\omega}$ of positive measure is simply the set ${Y_\omega^{X_\omega}}$ of functions from ${X_\omega}$ to ${Y_\omega}$.

In a similar spirit, we say that one stochastic set ${Y|\Omega}$ is a (stochastic) subset of another ${X|\Omega}$, and write ${Y|\Omega \subset X|\Omega}$, if we have a stochastic inclusion map, thus ${\Gamma(Y|E) \subset \Gamma(X|E)}$ for all events ${E}$, with the restriction maps being compatible. We can then define the power set ${2^X|\Omega}$ of a stochastic set ${X|\Omega}$ by setting ${\Gamma(2^X|E)}$ for any event ${E}$ to be the set of all stochastic subsets ${Y|E}$ of ${X|E}$ relative to ${E}$; it is easy to see that ${2^X|\Omega}$ is a stochastic set with the obvious restriction maps (one can also identify ${2^X|\Omega}$ with ${\hbox{Hom}(X, \{\hbox{true},\hbox{false}\})|\Omega}$ in the obvious fashion). Again, when ${\Omega}$ is discrete and at most countable, the fibre of ${2^X|\Omega}$ at a point ${\omega}$ of positive measure is simply the deterministic power set ${2^{X_\omega}}$.

Note that if ${f: X|\Omega \rightarrow Y|\Omega}$ is a stochastic function and ${Y'|\Omega}$ is a stochastic subset of ${Y|\Omega}$, then the inverse image ${f^{-1}(Y')|\Omega}$, defined by setting ${\Gamma(f^{-1}(Y')|E)}$ for any event ${E}$ to be the set of those ${x \in \Gamma(X|E)}$ with ${f(x) \in \Gamma(Y'|E)}$, is a stochastic subset of ${X|\Omega}$. In particular, given a ${k}$-ary relation ${R: X_1 \times \dots \times X_k|\Omega \rightarrow \{\hbox{true}, \hbox{false}\}|\Omega}$, the inverse image ${R^{-1}( \{ \hbox{true} \}|\Omega )}$ is a stochastic subset of ${X_1 \times \dots \times X_k|\Omega}$, which by abuse of notation we denote as

$\displaystyle \{ (x_1,\dots,x_k) \in X_1 \times \dots \times X_k: R(x_1,\dots,x_k) \hbox{ is true} \}|\Omega.$

In a similar spirit, if ${X'|\Omega}$ is a stochastic subset of ${X|\Omega}$ and ${f: X|\Omega \rightarrow Y|\Omega}$ is a stochastic function, we can define the image ${f(X')|\Omega}$ by setting ${\Gamma(f(X')|E)}$ to be the set of those ${f(x)}$ with ${x \in \Gamma(X'|E)}$; one easily verifies that this is a stochastic subset of ${Y|\Omega}$.

Remark 2 One should caution that in the definition of the subset relation ${Y|\Omega \subset X|\Omega}$, it is important that ${\Gamma(Y|E) \subset \Gamma(X|E)}$ for all events ${E}$, not just the global event ${\Omega}$; in particular, just because a stochastic set ${X|\Omega}$ has no global sections, does not mean that it is contained in the stochastic empty set ${\emptyset|\Omega}$.

Now we discuss Boolean operations on stochastic subsets of a given stochastic set ${X|\Omega}$. Given two stochastic subsets ${X_1|\Omega, X_2|\Omega}$ of ${X|\Omega}$, the stochastic intersection ${(X_1 \cap X_2)|\Omega}$ is defined by setting ${\Gamma((X_1 \cap X_2)|E)}$ to be the set of ${x \in \Gamma(X|E)}$ that lie in both ${\Gamma(X_1|E)}$ and ${\Gamma(X_2|E)}$:

$\displaystyle \Gamma(X_1 \cap X_2)|E) := \Gamma(X_1|E) \cap \Gamma(X_2|E).$

This is easily verified to again be a stochastic subset of ${X|\Omega}$. More generally one may define stochastic countable intersections ${(\bigcap_{n=1}^\infty X_n)|\Omega}$ for any sequence ${X_n|\Omega}$ of stochastic subsets of ${X|\Omega}$. One could extend this definition to uncountable families if one wished, but I would advise against it, because some of the usual laws of Boolean algebra (e.g. the de Morgan laws) may break down in this setting.

Stochastic unions are a bit more subtle. The set ${\Gamma((X_1 \cup X_2)|E)}$ should not be defined to simply be the union of ${\Gamma(X_1|E)}$ and ${\Gamma(X_2|E)}$, as this would not respect the gluing axiom. Instead, we define ${\Gamma((X_1 \cup X_2)|E)}$ to be the set of all ${x \in \Gamma(X|E)}$ such that one can cover ${E}$ by measurable subevents ${E_1,E_2}$ such that ${x_i|E_i \in \Gamma(X_i|E_i)}$ for ${i=1,2}$; then ${(X_1 \cup X_2)|\Omega}$ may be verified to be a stochastic subset of ${X|\Omega}$. Thus for instance ${\{0,1\}|\Omega}$ is the stochastic union of ${\{0\}|\Omega}$ and ${\{1\}|\Omega}$. Similarly for countable unions ${(\bigcup_{n=1}^\infty X_n)|\Omega}$ of stochastic subsets ${X_n|\Omega}$ of ${X|\Omega}$, although for uncountable unions are extremely problematic (they are disliked by both the measure theory and the countable gluing axiom) and will not be defined here. Finally, the stochastic difference set ${\Gamma((X_1 \backslash X_2)|E)}$ is defined as the set of all ${x|E}$ in ${\Gamma(X_1|E)}$ such that ${x|F \not \in \Gamma(X_2|F)}$ for any subevent ${F}$ of ${E}$ of positive probability. One may verify that in the case when ${\Omega}$ is discrete and at most countable, these Boolean operations correspond to the classical Boolean operations applied separately to each fibre ${X_{i,\omega}}$ of the relevant sets ${X_i}$. We also leave as an exercise to the reader to verify the usual laws of Boolean arithmetic, e.g. the de Morgan laws, provided that one works with at most countable unions and intersections.

One can also consider a stochastic finite union ${(\bigcup_{n=1}^N X_n)|\Omega}$ in which the number ${N}$ of sets in the union is itself stochastic. More precisely, let ${X|\Omega}$ be a stochastic set, let ${N \in {\bf N}|\Omega}$ be a stochastic natural number, and let ${n \mapsto X_n|\Omega}$ be a stochastic function from the stochastic set ${\{ n \in {\bf N}: n \leq N\}|\Omega}$ (defined by setting ${\Gamma(\{n \in {\bf N}: n\leq N\}|E) := \{ n \in {\bf N}|E: n \leq N|E\}}$)) to the stochastic power set ${2^X|\Omega}$. Here we are considering ${0}$ to be a natural number, to allow for unions that are possibly empty, with ${{\bf N}_+ := {\bf N} \backslash \{0\}}$ used for the positive natural numbers. We also write ${(X_n)_{n=1}^N|\Omega}$ for the stochastic function ${n \mapsto X_n|\Omega}$. Then we can define the stochastic union ${\bigcup_{n=1}^N X_n|\Omega}$ by setting ${\Gamma(\bigcup_{n=1}^N X_n|E)}$ for an event ${E}$ to be the set of local elements ${x \in \Gamma(X|E)}$ with the property that there exists a covering of ${E}$ by measurable subevents ${E_{n_0}}$ for ${n_0 \in {\bf N}_+}$, such that one has ${n_0 \leq N|E_{n_0}}$ and ${x|E_{n_0} \in \Gamma(X_{n_0}|E_{n_0})}$. One can verify that ${\bigcup_{n=1}^N X_n|\Omega}$ is a stochastic set (with the obvious restriction maps). Again, in the model case when ${\Omega}$ is discrete and at most countable, the fibre ${(\bigcup_{n=1}^N X_n)_\omega}$ is what one would expect it to be, namely ${\bigcup_{n=1}^{N(\omega)} (X_n)_\omega}$.

The Cartesian product ${(X \times Y)|\Omega}$ of two stochastic sets may be defined by setting ${\Gamma((X \times Y)|E) := \Gamma(X|E) \times \Gamma(Y|E)}$ for all events ${E}$, with the obvious restriction maps; this is easily seen to be another stochastic set. This lets one define the concept of a ${k}$-ary operation ${f: (X_1 \times \dots \times X_k)|\Omega \rightarrow Y|\Omega}$ from ${k}$ stochastic sets ${X_1,\dots,X_k}$ to another stochastic set ${Y}$, or a ${k}$-ary relation ${R: (X_1 \times \dots \times X_k)|\Omega \rightarrow \{\hbox{true}, \hbox{false}\}|\Omega}$. In particular, given ${x_i \in X_i|\Omega}$ for ${i=1,\dots,k}$, the relation ${R(x_1,\dots,x_k)}$ may be deterministically true, deterministically false, or have some other stochastic truth value.

Remark 3 In the degenerate case when ${\Omega}$ is null, stochastic logic becomes a bit weird: all stochastic statements are deterministically true, as are their stochastic negations, since every event in ${\Omega}$ (even the empty set) now holds with full probability. Among other pathologies, the empty set now has a global element over ${\Omega}$ (this is analogous to the notorious convention ${0^0=1}$), and any two deterministic objects ${x,y}$ become equal over ${\Omega}$: ${x|\Omega=y|\Omega}$.

The following simple observation is crucial to subsequent discussion. If ${(x_n)_{n \in {\bf N}_+}}$ is a sequence taking values in the global elements ${\Gamma(X|\Omega)}$ of a stochastic space ${X|\Omega}$, then we may also define global elements ${x_n \in \Gamma(X|\Omega)}$ for stochastic indices ${n \in {\bf N}_+|\Omega}$ as well, by appealing to the countable gluing axiom to glue together ${x_{n_0}}$ restricted to the set ${\{ \omega \in \Omega: n(\omega) = n_0\}}$ for each deterministic natural number ${n_0}$ to form ${x_n}$. With this definition, the map ${n \mapsto x_n}$ is a stochastic function from ${{\bf N}_+|\Omega}$ to ${X|\Omega}$; indeed, this creates a one-to-one correspondence between external sequences (maps ${n \mapsto x_n}$ from ${{\bf N}_+}$ to ${\Gamma(X|\Omega)}$) and stochastic sequences (stochastic functions ${n \mapsto x_n}$ from ${{\bf N}_+|\Omega}$ to ${X|\Omega}$). Similarly with ${{\bf N}_+}$ replaced by any other at most countable set. This observation will be important in allowing many deterministic arguments involving sequences will be able to be carried over to the stochastic setting.

We now specialise from the extremely broad discipline of set theory to the more focused discipline of real analysis. There are two fundamental axioms that underlie real analysis (and in particular distinguishes it from real algebra). The first is the Archimedean property, which we phrase in the “no infinitesimal” formulation as follows:

Proposition 2 (Archimedean property) Let ${x \in {\bf R}}$ be such that ${x \leq 1/n}$ for all positive natural numbers ${n}$. Then ${x \leq 0}$.

The other is the least upper bound axiom:

Proposition 3 (Least upper bound axiom) Let ${S}$ be a non-empty subset of ${{\bf R}}$ which has an upper bound ${M \in {\bf R}}$, thus ${x \leq M}$ for all ${x \in S}$. Then there exists a unique real number ${\sup S \in {\bf R}}$ with the following properties:

• ${x \leq \sup S}$ for all ${x \in S}$.
• For any real ${L < \sup S}$, there exists ${x \in S}$ such that ${L < x \leq \sup S}$.
• ${\sup S \leq M}$.

Furthermore, ${\sup S}$ does not depend on the choice of ${M}$.

The Archimedean property extends easily to the stochastic setting:

Proposition 4 (Stochastic Archimedean property) Let ${x \in \Gamma({\bf R}|\Omega)}$ be such that ${x \leq 1/n}$ for all deterministic natural numbers ${n}$. Then ${x \leq 0}$.

Remark 4 Here, incidentally, is one place in which this stochastic formalism deviates from the nonstandard analysis formalism, as the latter certainly permits the existence of infinitesimal elements. On the other hand, we caution that stochastic real numbers are permitted to be unbounded, so that formulation of Archimedean property is not valid in the stochastic setting.

The proof is easy and is left to the reader. The least upper bound axiom also extends nicely to the stochastic setting, but the proof requires more work (in particular, our argument uses the monotone convergence theorem):

Theorem 5 (Stochastic least upper bound axiom) Let ${S|\Omega}$ be a stochastic subset of ${{\bf R}|\Omega}$ which has a global upper bound ${M \in {\bf R}|\Omega}$, thus ${x \leq M}$ for all ${x \in \Gamma(S|\Omega)}$, and is globally non-empty in the sense that there is at least one global element ${x \in \Gamma(S|\Omega)}$. Then there exists a unique stochastic real number ${\sup S \in \Gamma({\bf R}|\Omega)}$ with the following properties:

• ${x \leq \sup S}$ for all ${x \in \Gamma(S|\Omega)}$.
• For any stochastic real ${L < \sup S}$, there exists ${x \in \Gamma(S|\Omega)}$ such that ${L < x \leq \sup S}$.
• ${\sup S \leq M}$.

Furthermore, ${\sup S}$ does not depend on the choice of ${M}$.

For future reference, we note that the same result holds with ${{\bf R}}$ replaced by ${{\bf N} \cup \{+\infty\}}$ throughout, since the latter may be embedded in the former, for instance by mapping ${n}$ to ${1 - \frac{1}{n+1}}$ and ${+\infty}$ to ${1}$. In applications, the above theorem serves as a reasonable substitute for the countable axiom of choice, which does not appear to hold in unrestricted generality relative to a measure space; in particular, it can be used to generate various extremising sequences for stochastic functionals on various stochastic function spaces.

Proof: Uniqueness is clear (using the Archimedean property), as well as the independence on ${M}$, so we turn to existence. By using an order-preserving map from ${{\bf R}}$ to ${(-1,1)}$ (e.g. ${x \mapsto \frac{2}{\pi} \hbox{arctan}(x)}$) we may assume that ${S|\Omega}$ is a subset of ${(-1,1)|\Omega}$, and that ${M < 1}$.

We observe that ${\Gamma(S|\Omega)}$ is a lattice: if ${x, y \in \Gamma(S|\Omega)}$, then ${\max(x,y)}$ and ${\min(x,y)}$ also lie in ${\Gamma(S|\Omega)}$. Indeed, ${\max(x,y)}$ may be formed by appealing to the countable gluing axiom to glue ${y}$ (restricted the set ${\{ \omega \in \Omega: x(\omega) < y(\omega) \}}$) with ${x}$ (restricted to the set ${\{ \omega \in \Omega: x(\omega) \geq y(\omega) \}}$), and similarly for ${\min(x,y)}$. (Here we use the fact that relations such as ${<}$ are Borel measurable on ${{\bf R}}$.)

Let ${A \in {\bf R}}$ denote the deterministic quantity

$\displaystyle A := \sup \{ \int_\Omega x(\omega)\ d\mu(\omega): x \in \Gamma(S|\Omega) \}$

then (by Proposition 3!) ${A}$ is well-defined; here we use the hypothesis that ${\mu(\Omega)}$ is finite. Thus we may find a sequence ${(x_n)_{n \in {\bf N}}}$ of elements ${x_n}$ of ${\Gamma(S|\Omega)}$ such that

$\displaystyle \int_\Omega x_n(\omega)\ d\mu(\omega) \rightarrow A \hbox{ as } n \rightarrow \infty. \ \ \ \ \ (1)$

Using the lattice property, we may assume that the ${x_n}$ are non-decreasing: ${x_n \leq x_m}$ whenever ${n \leq m}$. If we then define ${\sup S(\omega) := \sup_n x_n(\omega)}$ (after choosing measurable representatives of each equivalence class ${x_n}$), then ${\sup S}$ is a stochastic real with ${\sup S \leq M}$.

If ${x \in \Gamma(S|\Omega)}$, then ${\max(x,x_n) \in \Gamma(S|\Omega)}$, and so

$\displaystyle \int_\Omega \max(x,x_n)\ d\mu(\omega) \leq A.$

From this and (1) we conclude that

$\displaystyle \int_\Omega \max(x-x_n,0) \rightarrow 0 \hbox{ as } n \rightarrow \infty.$

From monotone convergence, we conclude that

$\displaystyle \int_\Omega \max(x-\sup S,0) = 0$

and so ${x \leq \sup S}$, as required.

Now let ${L < \sup S}$ be a stochastic real. After choosing measurable representatives of each relevant equivalence class, we see that for almost every ${\omega \in \Omega}$, we can find a natural number ${n(\omega)}$ with ${x_{n(\omega)} > L}$. If we choose ${n(\omega)}$ to be the first such positive natural number when it exists, and (say) ${1}$ otherwise, then ${n}$ is a stochastic positive natural number and ${L < x_n}$. The claim follows. $\Box$

Remark 5 One can abstract away the role of the measure ${\mu}$ here, leaving only the ideal of null sets. The property that the measure is finite is then replaced by the more general property that given any non-empty family of measurable sets, there is an at most countable union of sets in that family that is an upper bound modulo null sets for all elements in that faily.

Using Proposition 4 and Theorem 5, one can then revisit many of the other foundational results of deterministic real analysis, and develop stochastic analogues; we give some examples of this below the fold (focusing on the Heine-Borel theorem and a case of the spectral theorem). As an application of this formalism, we revisit some of the Furstenberg-Zimmer structural theory of measure-preserving systems, particularly that of relatively compact and relatively weakly mixing systems, and interpret them in this framework, basically as stochastic versions of compact and weakly mixing systems (though with the caveat that the shift map is allowed to act non-trivially on the underlying probability space). As this formalism is “point-free”, in that it avoids explicit use of fibres and disintegrations, it will be well suited for generalising this structure theory to settings in which the underlying probability spaces are not standard Borel, and the underlying groups are uncountable; I hope to discuss such generalisations in future blog posts.

Remark 6 Roughly speaking, stochastic real analysis can be viewed as a restricted subset of classical real analysis in which all operations have to be “measurable” with respect to the base space. In particular, indiscriminate application of the axiom of choice is not permitted, and one should largely restrict oneself to performing countable unions and intersections rather than arbitrary unions or intersections. Presumably one can formalise this intuition with a suitable “countable transfer principle”, but I was not able to formulate a clean and general principle of this sort, instead verifying various assertions about stochastic objects by hand rather than by direct transfer from the deterministic setting. However, it would be desirable to have such a principle, since otherwise one is faced with the tedious task of redoing all the foundations of real analysis (or whatever other base theory of mathematics one is going to be working in) in the stochastic setting by carefully repeating all the arguments.

More generally, topos theory is a good formalism for capturing precisely the informal idea of performing mathematics with certain operations, such as the axiom of choice, the law of the excluded middle, or arbitrary unions and intersections, being somehow “prohibited” or otherwise “restricted”.

Two of the most famous open problems in additive prime number theory are the twin prime conjecture and the binary Goldbach conjecture. They have quite similar forms:

• Twin prime conjecture The equation ${p_1 - p_2 = 2}$ has infinitely many solutions with ${p_1,p_2}$ prime.
• Binary Goldbach conjecture The equation ${p_1 + p_2 = N}$ has at least one solution with ${p_1,p_2}$ prime for any given even ${N \geq 4}$.

In view of this similarity, it is not surprising that the partial progress on these two conjectures have tracked each other fairly closely; the twin prime conjecture is generally considered slightly easier than the binary Goldbach conjecture, but broadly speaking any progress made on one of the conjectures has also led to a comparable amount of progress on the other. (For instance, Chen’s theorem has a version for the twin prime conjecture, and a version for the binary Goldbach conjecture.) Also, the notorious parity obstruction is present in both problems, preventing a solution to either conjecture by almost all known methods (see this previous blog post for more discussion).

In this post, I would like to note a divergence from this general principle, with regards to bounded error versions of these two conjectures:

• Twin prime with bounded error The inequalities ${0 < p_1 - p_2 < H}$ has infinitely many solutions with ${p_1,p_2}$ prime for some absolute constant ${H}$.
• Binary Goldbach with bounded error The inequalities ${N \leq p_1+p_2 \leq N+H}$ has at least one solution with ${p_1,p_2}$ prime for any sufficiently large ${N}$ and some absolute constant ${H}$.

The first of these statements is now a well-known theorem of Zhang, and the Polymath8b project hosted on this blog has managed to lower ${H}$ to ${H=246}$ unconditionally, and to ${H=6}$ assuming the generalised Elliott-Halberstam conjecture. However, the second statement remains open; the best result that the Polymath8b project could manage in this direction is that (assuming GEH) at least one of the binary Goldbach conjecture with bounded error, or the twin prime conjecture with no error, had to be true.

All the known proofs of Zhang’s theorem proceed through sieve-theoretic means. Basically, they take as input equidistribution results that control the size of discrepancies such as

$\displaystyle \Delta(f; a\ (q)) := \sum_{x \leq n \leq 2x; n=a\ (q)} f(n) - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x} f(n) \ \ \ \ \ (1)$

for various congruence classes ${a\ (q)}$ and various arithmetic functions ${f}$, e.g. ${f(n) = \Lambda(n+h_i)}$ (or more generaly ${f(n) = \alpha * \beta(n+h_i)}$ for various ${\alpha,\beta}$). After taking some carefully chosen linear combinations of these discrepancies, and using the trivial positivity lower bound

$\displaystyle a_n \geq 0 \hbox{ for all } n \implies \sum_n a_n \geq 0 \ \ \ \ \ (2)$

one eventually obtains (for suitable ${H}$) a non-trivial lower bound of the form

$\displaystyle \sum_{x \leq n \leq 2x} \nu(n) 1_A(n) > 0$

where ${\nu}$ is some weight function, and ${A}$ is the set of ${n}$ such that there are at least two primes in the interval ${[n,n+H]}$. This implies at least one solution to the inequalities ${0 < p_1 - p_2 < H}$ with ${p_1,p_2 \sim x}$, and Zhang’s theorem follows.

In a similar vein, one could hope to use bounds on discrepancies such as (1) (for ${x}$ comparable to ${N}$), together with the trivial lower bound (2), to obtain (for sufficiently large ${N}$, and suitable ${H}$) a non-trivial lower bound of the form

$\displaystyle \sum_{n \leq N} \nu(n) 1_B(n) > 0 \ \ \ \ \ (3)$

for some weight function ${\nu}$, where ${B}$ is the set of ${n}$ such that there is at least one prime in each of the intervals ${[n,n+H]}$ and ${[N-n-H,n]}$. This would imply the binary Goldbach conjecture with bounded error.

However, the parity obstruction blocks such a strategy from working (for much the same reason that it blocks any bound of the form ${H \leq 4}$ in Zhang’s theorem, as discussed in the Polymath8b paper.) The reason is as follows. The sieve-theoretic arguments are linear with respect to the ${n}$ summation, and as such, any such sieve-theoretic argument would automatically also work in a weighted setting in which the ${n}$ summation is weighted by some non-negative weight ${\omega(n) \geq 0}$. More precisely, if one could control the weighted discrepancies

$\displaystyle \Delta(f\omega; a\ (q)) = \sum_{x \leq n \leq 2x; n=a\ (q)} f(n) \omega(n) - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x} f(n) \omega(n)$

to essentially the same accuracy as the unweighted discrepancies (1), then thanks to the trivial weighted version

$\displaystyle a_n \geq 0 \hbox{ for all } n \implies \sum_n a_n \omega(n) \geq 0$

of (2), any sieve-theoretic argument that was capable of proving (3) would also be capable of proving the weighted estimate

$\displaystyle \sum_{n \leq N} \nu(n) 1_B(n) \omega(n) > 0. \ \ \ \ \ (4)$

However, (4) may be defeated by a suitable choice of weight ${\omega}$, namely

$\displaystyle \omega(n) := \prod_{i=1}^H (1 + \lambda(n) \lambda(n+i)) \times \prod_{j=0}^H (1 - \lambda(n) \lambda(N-n-j))$

where ${n \mapsto \lambda(n)}$ is the Liouville function, which counts the parity of the number of prime factors of a given number ${n}$. Since ${\lambda(n)^2 = 1}$, one can expand out ${\omega(n)}$ as the sum of ${1}$ and a finite number of other terms, each of which consists of the product of two or more translates (or reflections) of ${\lambda}$. But from the Möbius randomness principle (or its analogue for the Liouville function), such products of ${\lambda}$ are widely expected to be essentially orthogonal to any arithmetic function ${f(n)}$ that is arising from a single multiplicative function such as ${\Lambda}$, even on very short arithmetic progressions. As such, replacing ${1}$ by ${\omega(n)}$ in (1) should have a negligible effect on the discrepancy. On the other hand, in order for ${\omega(n)}$ to be non-zero, ${\lambda(n+i)}$ has to have the same sign as ${\lambda(n)}$ and hence the opposite sign to ${\lambda(N-n-j)}$ cannot simultaneously be prime for any ${0 \leq i,j \leq H}$, and so ${1_B(n) \omega(n)}$ vanishes identically, contradicting (4). This indirectly rules out any modification of the Goldston-Pintz-Yildirim/Zhang method for establishing the binary Goldbach conjecture with bounded error.

The above argument is not watertight, and one could envisage some ways around this problem. One of them is that the Möbius randomness principle could simply be false, in which case the parity obstruction vanishes. A good example of this is the result of Heath-Brown that shows that if there are infinitely many Siegel zeroes (which is a strong violation of the Möbius randomness principle), then the twin prime conjecture holds. Another way around the obstruction is to start controlling the discrepancy (1) for functions ${f}$ that are combinations of more than one multiplicative function, e.g. ${f(n) = \Lambda(n) \Lambda(n+2)}$. However, controlling such functions looks to be at least as difficult as the twin prime conjecture (which is morally equivalent to obtaining non-trivial lower-bounds for ${\sum_{x \leq n \leq 2x} \Lambda(n) \Lambda(n+2)}$). A third option is not to use a sieve-theoretic argument, but to try a different method (e.g. the circle method). However, most other known methods also exhibit linearity in the “${n}$” variable and I would suspect they would be vulnerable to a similar obstruction. (In any case, the circle method specifically has some other difficulties in tackling binary problems, as discussed in this previous post.)

Let ${\bar{{\bf Q}}}$ be the algebraic closure of ${{\bf Q}}$, that is to say the field of algebraic numbers. We fix an embedding of ${\bar{{\bf Q}}}$ into ${{\bf C}}$, giving rise to a complex absolute value ${z \mapsto |z|}$ for algebraic numbers ${z \in \bar{{\bf Q}}}$.

Let ${\alpha \in \bar{{\bf Q}}}$ be of degree ${D > 1}$, so that ${\alpha}$ is irrational. A classical theorem of Liouville gives the quantitative bound

$\displaystyle |\alpha - \frac{p}{q}| \geq c \frac{1}{|q|^D} \ \ \ \ \ (1)$

for the irrationality of ${\alpha}$ fails to be approximated by rational numbers ${p/q}$, where ${c>0}$ depends on ${\alpha,D}$ but not on ${p,q}$. Indeed, if one lets ${\alpha = \alpha_1, \alpha_2, \dots, \alpha_D}$ be the Galois conjugates of ${\alpha}$, then the quantity ${\prod_{i=1}^D |q \alpha_i - p|}$ is a non-zero natural number divided by a constant, and so we have the trivial lower bound

$\displaystyle \prod_{i=1}^D |q \alpha_i - p| \geq c$

from which the bound (1) easily follows. A well known corollary of the bound (1) is that Liouville numbers are automatically transcendental.

The famous theorem of Thue, Siegel and Roth improves the bound (1) to

$\displaystyle |\alpha - \frac{p}{q}| \geq c \frac{1}{|q|^{2+\epsilon}} \ \ \ \ \ (2)$

for any ${\epsilon>0}$ and rationals ${\frac{p}{q}}$, where ${c>0}$ depends on ${\alpha,\epsilon}$ but not on ${p,q}$. Apart from the ${\epsilon}$ in the exponent and the implied constant, this bound is optimal, as can be seen from Dirichlet’s theorem. This theorem is a good example of the ineffectivity phenomenon that affects a large portion of modern number theory: the implied constant in the ${\gg}$ notation is known to be finite, but there is no explicit bound for it in terms of the coefficients of the polynomial defining ${\alpha}$ (in contrast to (1), for which an effective bound may be easily established). This is ultimately due to the reliance on the “dueling conspiracy” (or “repulsion phenomenon”) strategy. We do not as yet have a good way to rule out one counterexample to (2), in which ${\frac{p}{q}}$ is far closer to ${\alpha}$ than ${\frac{1}{|q|^{2+\epsilon}}}$; however we can rule out two such counterexamples, by playing them off of each other.

A powerful strengthening of the Thue-Siegel-Roth theorem is given by the subspace theorem, first proven by Schmidt and then generalised further by several authors. To motivate the theorem, first observe that the Thue-Siegel-Roth theorem may be rephrased as a bound of the form

$\displaystyle | \alpha p - \beta q | \times | \alpha' p - \beta' q | \geq c (1 + |p| + |q|)^{-\epsilon} \ \ \ \ \ (3)$

for any algebraic numbers ${\alpha,\beta,\alpha',\beta'}$ with ${(\alpha,\beta)}$ and ${(\alpha',\beta')}$ linearly independent (over the algebraic numbers), and any ${(p,q) \in {\bf Z}^2}$ and ${\epsilon>0}$, with the exception when ${\alpha,\beta}$ or ${\alpha',\beta'}$ are rationally dependent (i.e. one is a rational multiple of the other), in which case one has to remove some lines (i.e. subspaces in ${{\bf Q}^2}$) of rational slope from the space ${{\bf Z}^2}$ of pairs ${(p,q)}$ to which the bound (3) does not apply (namely, those lines for which the left-hand side vanishes). Here ${c>0}$ can depend on ${\alpha,\beta,\alpha',\beta',\epsilon}$ but not on ${p,q}$. More generally, we have

Theorem 1 (Schmidt subspace theorem) Let ${d}$ be a natural number. Let ${L_1,\dots,L_d: \bar{{\bf Q}}^d \rightarrow \bar{{\bf Q}}}$ be linearly independent linear forms. Then for any ${\epsilon>0}$, one has the bound

$\displaystyle \prod_{i=1}^d |L_i(x)| \geq c (1 + \|x\| )^{-\epsilon}$

for all ${x \in {\bf Z}^d}$, outside of a finite number of proper subspaces of ${{\bf Q}^d}$, where

$\displaystyle \| (x_1,\dots,x_d) \| := \max( |x_1|, \dots, |x_d| )$

and ${c>0}$ depends on ${\epsilon, d}$ and the ${\alpha_{i,j}}$, but is independent of ${x}$.

Being a generalisation of the Thue-Siegel-Roth theorem, it is unsurprising that the known proofs of the subspace theorem are also ineffective with regards to the constant ${c}$. (However, the number of exceptional subspaces may be bounded effectively; cf. the situation with the Skolem-Mahler-Lech theorem, discussed in this previous blog post.) Once again, the lower bound here is basically sharp except for the ${\epsilon}$ factor and the implied constant: given any ${\delta_1,\dots,\delta_d > 0}$ with ${\delta_1 \dots \delta_d = 1}$, a simple volume packing argument (the same one used to prove the Dirichlet approximation theorem) shows that for any sufficiently large ${N \geq 1}$, one can find integers ${x_1,\dots,x_d \in [-N,N]}$, not all zero, such that

$\displaystyle |L_i(x)| \ll \delta_i$

for all ${i=1,\dots,d}$. Thus one can get ${\prod_{i=1}^d |L_i(x)|}$ comparable to ${1}$ in many different ways.

There are important generalisations of the subspace theorem to other number fields than the rationals (and to other valuations than the Archimedean valuation ${z \mapsto |z|}$); we will develop one such generalisation below.

The subspace theorem is one of many finiteness theorems in Diophantine geometry; in this case, it is the number of exceptional subspaces which is finite. It turns out that finiteness theorems are very compatible with the language of nonstandard analysis. (See this previous blog post for a review of the basics of nonstandard analysis, and in particular for the nonstandard interpretation of asymptotic notation such as ${\ll}$ and ${o()}$.) The reason for this is that a standard set ${X}$ is finite if and only if it contains no strictly nonstandard elements (that is to say, elements of ${{}^* X \backslash X}$). This makes for a clean formulation of finiteness theorems in the nonstandard setting. For instance, the standard form of Bezout’s theorem asserts that if ${P(x,y), Q(x,y)}$ are coprime polynomials over some field, then the curves ${\{ (x,y): P(x,y) = 0\}}$ and ${\{ (x,y): Q(x,y)=0\}}$ intersect in only finitely many points. The nonstandard version of this is then

Theorem 2 (Bezout’s theorem, nonstandard form) Let ${P(x,y), Q(x,y)}$ be standard coprime polynomials. Then there are no strictly nonstandard solutions to ${P(x,y)=Q(x,y)=0}$.

Now we reformulate Theorem 1 in nonstandard language. We need a definition:

Definition 3 (General position) Let ${K \subset L}$ be nested fields. A point ${x = (x_1,\dots,x_d)}$ in ${L^d}$ is said to be in ${K}$-general position if it is not contained in any hyperplane of ${L^d}$ definable over ${K}$, or equivalently if one has

$\displaystyle a_1 x_1 + \dots + a_d x_d = 0 \iff a_1=\dots = a_d = 0$

for any ${a_1,\dots,a_d \in K}$.

Theorem 4 (Schmidt subspace theorem, nonstandard version) Let ${d}$ be a standard natural number. Let ${L_1,\dots,L_d: \bar{{\bf Q}}^d \rightarrow \bar{{\bf Q}}}$ be linearly independent standard linear forms. Let ${x \in {}^* {\bf Z}^d}$ be a tuple of nonstandard integers which is in ${{\bf Q}}$-general position (in particular, this forces ${x}$ to be strictly nonstandard). Then one has

$\displaystyle \prod_{i=1}^d |L_i(x)| \gg \|x\|^{-o(1)},$

where we extend ${L_i}$ from ${\bar{{\bf Q}}}$ to ${{}^* \bar{{\bf Q}}}$ (and also similarly extend ${\| \|}$ from ${{\bf Z}^d}$ to ${{}^* {\bf Z}^d}$) in the usual fashion.

Observe that (as is usual when translating to nonstandard analysis) some of the epsilons and quantifiers that are present in the standard version become hidden in the nonstandard framework, being moved inside concepts such as “strictly nonstandard” or “general position”. We remark that as ${x}$ is in ${{\bf Q}}$-general position, it is also in ${\bar{{\bf Q}}}$-general position (as an easy Galois-theoretic argument shows), and the requirement that the ${L_1,\dots,L_d}$ are linearly independent is thus equivalent to ${L_1(x),\dots,L_d(x)}$ being ${\bar{{\bf Q}}}$-linearly independent.

Exercise 1 Verify that Theorem 1 and Theorem 4 are equivalent. (Hint: there are only countably many proper subspaces of ${{\bf Q}^d}$.)

We will not prove the subspace theorem here, but instead focus on a particular application of the subspace theorem, namely to counting integer points on curves. In this paper of Corvaja and Zannier, the subspace theorem was used to give a new proof of the following basic result of Siegel:

Theorem 5 (Siegel’s theorem on integer points) Let ${P \in {\bf Q}[x,y]}$ be an irreducible polynomial of two variables, such that the affine plane curve ${C := \{ (x,y): P(x,y)=0\}}$ either has genus at least one, or has at least three points on the line at infinity, or both. Then ${C}$ has only finitely many integer points ${(x,y) \in {\bf Z}^2}$.

This is a finiteness theorem, and as such may be easily converted to a nonstandard form:

Theorem 6 (Siegel’s theorem, nonstandard form) Let ${P \in {\bf Q}[x,y]}$ be a standard irreducible polynomial of two variables, such that the affine plane curve ${C := \{ (x,y): P(x,y)=0\}}$ either has genus at least one, or has at least three points on the line at infinity, or both. Then ${C}$ does not contain any strictly nonstandard integer points ${(x_*,y_*) \in {}^* {\bf Z}^2 \backslash {\bf Z}^2}$.

Note that Siegel’s theorem can fail for genus zero curves that only meet the line at infinity at just one or two points; the key examples here are the graphs ${\{ (x,y): y - f(x) = 0\}}$ for a polynomial ${f \in {\bf Z}[x]}$, and the Pell equation curves ${\{ (x,y): x^2 - dy^2 = 1 \}}$. Siegel’s theorem can be compared with the more difficult theorem of Faltings, which establishes finiteness of rational points (not just integer points), but now needs the stricter requirement that the curve ${C}$ has genus at least two (to avoid the additional counterexample of elliptic curves of positive rank, which have infinitely many rational points).

The standard proofs of Siegel’s theorem rely on a combination of the Thue-Siegel-Roth theorem and a number of results on abelian varieties (notably the Mordell-Weil theorem). The Corvaja-Zannier argument rebalances the difficulty of the argument by replacing the Thue-Siegel-Roth theorem by the more powerful subspace theorem (in fact, they need one of the stronger versions of this theorem alluded to earlier), while greatly reducing the reliance on results on abelian varieties. Indeed, for curves with three or more points at infinity, no theory from abelian varieties is needed at all, while for the remaining cases, one mainly needs the existence of the Abel-Jacobi embedding, together with a relatively elementary theorem of Chevalley-Weil which is used in the proof of the Mordell-Weil theorem, but is significantly easier to prove.

The Corvaja-Zannier argument (together with several further applications of the subspace theorem) is presented nicely in this Bourbaki expose of Bilu. To establish the theorem in full generality requires a certain amount of algebraic number theory machinery, such as the theory of valuations on number fields, or of relative discriminants between such number fields. However, the basic ideas can be presented without much of this machinery by focusing on simple special cases of Siegel’s theorem. For instance, we can handle irreducible cubics that meet the line at infinity at exactly three points ${[1,\alpha_1,0], [1,\alpha_2,0], [1,\alpha_3,0]}$:

Theorem 7 (Siegel’s theorem with three points at infinity) Siegel’s theorem holds when the irreducible polynomial ${P(x,y)}$ takes the form

$\displaystyle P(x,y) = (y - \alpha_1 x) (y - \alpha_2 x) (y - \alpha_3 x) + Q(x,y)$

for some quadratic polynomial ${Q \in {\bf Q}[x,y]}$ and some distinct algebraic numbers ${\alpha_1,\alpha_2,\alpha_3}$.

Proof: We use the nonstandard formalism. Suppose for sake of contradiction that we can find a strictly nonstandard integer point ${(x_*,y_*) \in {}^* {\bf Z}^2 \backslash {\bf Z}^2}$ on a curve ${C := \{ (x,y): P(x,y)=0\}}$ of the indicated form. As this point is infinitesimally close to the line at infinity, ${y_*/x_*}$ must be infinitesimally close to one of ${\alpha_1,\alpha_2,\alpha_3}$; without loss of generality we may assume that ${y_*/x_*}$ is infinitesimally close to ${\alpha_1}$.

We now use a version of the polynomial method, to find some polynomials of controlled degree that vanish to high order on the “arm” of the cubic curve ${C}$ that asymptotes to ${[1,\alpha_1,0]}$. More precisely, let ${D \geq 3}$ be a large integer (actually ${D=3}$ will already suffice here), and consider the ${\bar{{\bf Q}}}$-vector space ${V}$ of polynomials ${R(x,y) \in \bar{{\bf Q}}[x,y]}$ of degree at most ${D}$, and of degree at most ${2}$ in the ${y}$ variable; this space has dimension ${3D}$. Also, as one traverses the arm ${y/x \rightarrow \alpha_1}$ of ${C}$, any polynomial ${R}$ in ${V}$ grows at a rate of at most ${D}$, that is to say ${R}$ has a pole of order at most ${D}$ at the point at infinity ${[1,\alpha_1,0]}$. By performing Laurent expansions around this point (which is a non-singular point of ${C}$, as the ${\alpha_i}$ are assumed to be distinct), we may thus find a basis ${R_1, \dots, R_{3D}}$ of ${V}$, with the property that ${R_j}$ has a pole of order at most ${D+1-j}$ at ${[1,\alpha_1,0]}$ for each ${j=1,\dots,3D}$.

From the control of the pole at ${[1,\alpha_1,0]}$, we have

$\displaystyle |R_j(x_*,y_*)| \ll (|x_*|+|y_*|)^{D+1-j}$

for all ${j=1,\dots,3D}$. The exponents here become negative for ${j > D+1}$, and on multiplying them all together we see that

$\displaystyle \prod_{j=1}^{3D} |R_j(x_*,y_*)| \ll (|x_*|+|y_*|)^{3D(D+1) - \frac{3D(3D+1)}{2}}.$

This exponent is negative for ${D}$ large enough (or just take ${D=3}$). If we expand

$\displaystyle R_j(x_*,y_*) = \sum_{a+b \leq D; b \leq 2} \alpha_{j,a,b} x_*^a y_*^b$

for some algebraic numbers ${\alpha_{j,a,b}}$, then we thus have

$\displaystyle \prod_{j=1}^{3D} |\sum_{a+b \leq D; b \leq 2} \alpha_{j,a,b} x_*^a y_*^b| \ll (|x_*|+|y_*|)^{-\epsilon}$

for some standard ${\epsilon>0}$. Note that the ${3D}$-dimensional vectors ${(\alpha_{j,a,b})_{a+b \leq D; b \leq 2}}$ are linearly independent in ${{\bf C}^{3D}}$, because the ${R_j}$ are linearly independent in ${V}$. Applying the Schmidt subspace theorem in the contrapositive, we conclude that the ${3D}$-tuple ${( x_*^a y_*^b )_{a+b \leq D; b \leq 2} \in {}^* {\bf Z}^{3D}}$ is not in ${{\bf Q}}$-general position. That is to say, one has a non-trivial constraint of the form

$\displaystyle \sum_{a+b \leq D; b \leq 2} c_{a,b} x_*^a y_*^b = 0 \ \ \ \ \ (4)$

for some standard rational coefficients ${c_{a,b}}$, not all zero. But, as ${P}$ is irreducible and cubic in ${y}$, it has no common factor with the standard polynomial ${\sum_{a+b \leq D; b \leq 2} c_{a,b} x^a y^b}$, so by Bezout’s theorem (Theorem 2) the constraint (4) only has standard solutions, contradicting the strictly nonstandard nature of ${(x_*,y_*)}$. $\Box$

Exercise 2 Rewrite the above argument so that it makes no reference to nonstandard analysis. (In this case, the rewriting is quite straightforward; however, there will be a subsequent argument in which the standard version is significantly messier than the nonstandard counterpart, which is the reason why I am working with the nonstandard formalism in this blog post.)

A similar argument works for higher degree curves that meet the line at infinity in three or more points, though if the curve has singularities at infinity then it becomes convenient to rely on the Riemann-Roch theorem to control the dimension of the analogue of the space ${V}$. Note that when there are only two or fewer points at infinity, though, one cannot get the negative exponent of ${-\epsilon}$ needed to usefully apply the subspace theorem. To deal with this case we require some additional tricks. For simplicity we focus on the case of Mordell curves, although it will be convenient to work with more general number fields ${{\bf Q} \subset K \subset \bar{{\bf Q}}}$ than the rationals:

Theorem 8 (Siegel’s theorem for Mordell curves) Let ${k}$ be a non-zero integer. Then there are only finitely many integer solutions ${(x,y) \in {\bf Z}^2}$ to ${y^2 - x^3 = k}$. More generally, for any number field ${K}$, and any nonzero ${k \in K}$, there are only finitely many algebraic integer solutions ${(x,y) \in {\mathcal O}_K^2}$ to ${y^2-x^3=k}$, where ${{\mathcal O}_K}$ is the ring of algebraic integers in ${K}$.

Again, we will establish the nonstandard version. We need some additional notation:

Definition 9

• We define an almost rational integer to be a nonstandard ${x \in {}^* {\bf Q}}$ such that ${Mx \in {}^* {\bf Z}}$ for some standard positive integer ${M}$, and write ${{\bf Q} {}^* {\bf Z}}$ for the ${{\bf Q}}$-algebra of almost rational integers.
• If ${K}$ is a standard number field, we define an almost ${K}$-integer to be a nonstandard ${x \in {}^* K}$ such that ${Mx \in {}^* {\mathcal O}_K}$ for some standard positive integer ${M}$, and write ${K {}^* {\bf Z} = K {\mathcal O}_K}$ for the ${K}$-algebra of almost ${K}$-integers.
• We define an almost algebraic integer to be a nonstandard ${x \in {}^* {\bar Q}}$ such that ${Mx}$ is a nonstandard algebraic integer for some standard positive integer ${M}$, and write ${\bar{{\bf Q}} {}^* {\bf Z}}$ for the ${\bar{{\bf Q}}}$-algebra of almost algebraic integers.
• Theorem 10 (Siegel for Mordell, nonstandard version) Let ${k}$ be a non-zero standard algebraic number. Then the curve ${\{ (x,y): y^2 - x^3 = k \}}$ does not contain any strictly nonstandard almost algebraic integer point.

Another way of phrasing this theorem is that if ${x,y}$ are strictly nonstandard almost algebraic integers, then ${y^2-x^3}$ is either strictly nonstandard or zero.

Exercise 3 Verify that Theorem 8 and Theorem 10 are equivalent.

Due to all the ineffectivity, our proof does not supply any bound on the solutions ${x,y}$ in terms of ${k}$, even if one removes all references to nonstandard analysis. It is a conjecture of Hall (a special case of the notorious ABC conjecture) that one has the bound ${|x| \ll_\epsilon |k|^{2+\epsilon}}$ for all ${\epsilon>0}$ (or equivalently ${|y| \ll_\epsilon |k|^{3+\epsilon}}$), but even the weaker conjecture that ${x,y}$ are of polynomial size in ${k}$ is open. (The best known bounds are of exponential nature, and are proven using a version of Baker’s method: see for instance this text of Sprindzuk.)

A direct repetition of the arguments used to prove Theorem 7 will not work here, because the Mordell curve ${\{ (x,y): y^2 - x^3 = k \}}$ only hits the line at infinity at one point, ${[0,1,0]}$. To get around this we will exploit the fact that the Mordell curve is an elliptic curve and thus has a group law on it. We will then divide all the integer points on this curve by two; as elliptic curves have four 2-torsion points, this will end up placing us in a situation like Theorem 7, with four points at infinity. However, there is an obstruction: it is not obvious that dividing an integer point on the Mordell curve by two will produce another integer point. However, this is essentially true (after enlarging the ring of integers slightly) thanks to a general principle of Chevalley and Weil, which can be worked out explicitly in the case of division by two on Mordell curves by relatively elementary means (relying mostly on unique factorisation of ideals of algebraic integers). We give the details below the fold.

As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet ${(X, {\mathcal X}, \mu)}$, where ${X}$ is a set, ${{\mathcal X}}$ is a ${\sigma}$-algebra of subsets of ${X}$, and ${\mu: {\mathcal X} \rightarrow [0,1]}$ is a countably additive probability measure on ${{\mathcal X}}$. Given such a space, one can form a number of interesting function spaces, including

• the (real) Hilbert space ${L^2(X, {\mathcal X}, \mu)}$ of square-integrable functions ${f: X \rightarrow {\bf R}}$, modulo ${\mu}$-almost everywhere equivalence, and with the positive definite inner product ${\langle f, g\rangle_{L^2(X, {\mathcal X}, \mu)} := \int_X f g\ d\mu}$; and
• the unital commutative Banach algebra ${L^\infty(X, {\mathcal X}, \mu)}$ of essentially bounded functions ${f: X \rightarrow {\bf R}}$, modulo ${\mu}$-almost everywhere equivalence, with ${\|f\|_{L^\infty(X, {\mathcal X}, \mu)}}$ defined as the essential supremum of ${|f|}$.

There is also a trace ${\tau = \tau_\mu: L^\infty(X, {\mathcal X}, \mu) \rightarrow {\bf C}}$ on ${L^\infty}$ defined by integration: ${\tau(f) := \int_X f\ d\mu}$.

One can form the category ${\mathbf{Prb}}$ of classical probability spaces, by defining a morphism ${\phi: (X, {\mathcal X}, \mu) \rightarrow (Y, {\mathcal Y}, \nu)}$ between probability spaces to be a function ${\phi: X \rightarrow Y}$ which is measurable (thus ${\phi^{-1}(E) \in {\mathcal X}}$ for all ${E \in {\mathcal Y}}$) and measure-preserving (thus ${\mu(\phi^{-1}(E)) = \nu(E)}$ for all ${E \in {\mathcal Y}}$).

Let us now abstract the algebraic features of these spaces as follows; for want of a better name, I will refer to this abstraction as an algebraic probability space, and is very similar to the non-commutative probability spaces studied in this previous post, except that these spaces are now commutative (and real).

Definition 1 An algebraic probability space is a pair ${({\mathcal A}, \tau)}$ where

• ${{\mathcal A}}$ is a unital commutative real algebra;
• ${\tau: {\mathcal A} \rightarrow {\bf R}}$ is a homomorphism such that ${\tau(1)=1}$ and ${\tau( f^2 ) \geq 0}$ for all ${f \in {\mathcal A}}$;
• Every element ${f}$ of ${{\mathcal A}}$ is bounded in the sense that ${\sup_{k \geq 1} \tau( f^{2k} )^{1/2k} < \infty}$. (Technically, this isn’t an algebraic property, but I need it for technical reasons.)

A morphism ${\phi: ({\mathcal A}_1, \tau_1) \rightarrow ({\mathcal A}_2, \tau_2)}$ is a homomorphism ${\phi^*: {\mathcal A}_2 \rightarrow {\mathcal A}_1}$ which is trace-preserving, in the sense that ${\tau_1(\phi^*(f)) = \tau_2(f)}$ for all ${f \in {\mathcal A}_2}$.

For want of a better name, I’ll denote the category of algebraic probability spaces as ${\mathbf{AlgPrb}}$. One can view this category as the opposite category to that of (a subcategory of) the category of tracial commutative real algebras. One could emphasise this opposite nature by denoting the algebraic probability space as ${({\mathcal A}, \tau)^{op}}$ rather than ${({\mathcal A},\tau)}$; another suggestive (but slightly inaccurate) notation, inspired by the language of schemes, would be ${\hbox{Spec}({\mathcal A},\tau)}$ rather than ${({\mathcal A},\tau)}$. However, we will not adopt these conventions here, and refer to algebraic probability spaces just by the pair ${({\mathcal A},\tau)}$.

By the previous discussion, we have a covariant functor ${F: \textbf{Prb} \rightarrow \textbf{AlgPrb}}$ that takes a classical probability space ${(X, {\mathcal X}, \mu)}$ to its algebraic counterpart ${(L^\infty(X, {\mathcal X},\mu), \tau_\mu)}$, with a morphism ${\phi: (X, {\mathcal X}, \mu) \rightarrow (Y, {\mathcal Y}, \nu)}$ of classical probability spaces mapping to a morphism ${F(\phi): (L^\infty(X, {\mathcal X},\mu), \tau_\mu) \rightarrow (L^\infty(Y, {\mathcal Y},\nu), \tau_\nu)}$ of the corresponding algebraic probability spaces by the formula

$\displaystyle F(\phi)^* f := f \circ \phi$

for ${f \in L^\infty(Y, {\mathcal Y}, \nu)}$. One easily verifies that this is a functor.

In this post I would like to describe a functor ${G: \textbf{AlgPrb} \rightarrow \textbf{Prb}}$ which partially inverts ${F}$ (up to natural isomorphism), that is to say a recipe for starting with an algebraic probability space ${({\mathcal A}, \tau)}$ and producing a classical probability space ${(X, {\mathcal X}, \mu)}$. This recipe is not new – it is basically the (commutative) Gelfand-Naimark-Segal construction (discussed in this previous post) combined with the Loomis-Sikorski theorem (discussed in this previous post). However, I wanted to put the construction in a single location for sake of reference. I also wanted to make the point that ${F}$ and ${G}$ are not complete inverses; there is a bit of information in the algebraic probability space (e.g. topological information) which is lost when passing back to the classical probability space. In some future posts, I would like to develop some ergodic theory using the algebraic foundations of probability theory rather than the classical foundations; this turns out to be convenient in the ergodic theory arising from nonstandard analysis (such as that described in this previous post), in which the groups involved are uncountable and the underlying spaces are not standard Borel spaces.

Let us describe how to construct the functor ${G}$, with details postponed to below the fold.

1. Starting with an algebraic probability space ${({\mathcal A}, \tau)}$, form an inner product on ${{\mathcal A}}$ by the formula ${\langle f, g \rangle := \tau(fg)}$, and also form the spectral radius ${\rho(f) :=\lim_{k \rightarrow \infty} \tau(f^{2^k})^{1/2^k}}$.
2. The inner product is clearly positive semi-definite. Quotienting out the null vectors and taking completions, we arrive at a real Hilbert space ${L^2 = L^2({\mathcal A},\tau)}$, to which the trace ${\tau}$ may be extended.
3. Somewhat less obviously, the spectral radius is well-defined and gives a norm on ${{\mathcal A}}$. Taking ${L^2}$ limits of sequences in ${{\mathcal A}}$ of bounded spectral radius gives us a subspace ${L^\infty = L^\infty({\mathcal A},\tau)}$ of ${L^2}$ that has the structure of a real commutative Banach algebra.
4. The idempotents ${1_E}$ of the Banach algebra ${L^\infty}$ may be indexed by elements ${E}$ of an abstract ${\sigma}$-algebra ${{\mathcal B}}$.
5. The Boolean algebra homomorphisms ${\delta_x: {\mathcal B} \rightarrow \{0,1\}}$ (or equivalently, the real algebra homomorphisms ${\iota_x: L^\infty \rightarrow {\bf R}}$) may be indexed by elements ${x}$ of a space ${X}$.
6. Let ${{\mathcal X}}$ denote the ${\sigma}$-algebra on ${X}$ generated by the basic sets ${\overline{E} := \{ x \in X: \delta_x(E) = 1 \}}$ for every ${E \in {\mathcal B}}$.
7. Let ${{\mathcal N}}$ be the ${\sigma}$-ideal of ${{\mathcal X}}$ generated by the sets ${\bigcap_n \overline{E_n}}$, where ${E_n \in {\mathcal B}}$ is a sequence with ${\bigcap_n E_n = \emptyset}$.
8. One verifies that ${{\mathcal B}}$ is isomorphic to ${{\mathcal X}/{\mathcal N}}$. Using this isomorphism, the trace ${\tau}$ on ${L^\infty}$ can be used to construct a countably additive measure ${\mu}$ on ${{\mathcal X}}$. The classical probability space ${(X, {\mathcal X}, \mu)}$ is then ${G( {\mathcal A}, \tau )}$, and the abstract spaces ${L^2, L^\infty}$ may now be identified with their concrete counterparts ${L^2(X, {\mathcal X}, \mu)}$, ${L^\infty(X, {\mathcal X}, \mu)}$.
9. Every algebraic probability space morphism ${\phi: ({\mathcal A}_1,\tau_1) \rightarrow ({\mathcal A}_2,\tau_2)}$ generates a classical probability morphism ${G(\phi): (X_1, {\mathcal X}_1, \mu_1) \rightarrow (X_2, {\mathcal X}_2, \mu_2)}$ via the formula

$\displaystyle \delta_{G(\phi)(x_1)}( E_2 ) = \delta_{x_1}( \phi^*(E_2) )$

using a pullback operation ${\phi^*}$ on the abstract ${\sigma}$-algebras ${{\mathcal B}_1, {\mathcal B}_2}$ that can be defined by density.

Remark 1 The classical probability space ${X}$ constructed by the functor ${G}$ has some additional structure; namely ${X}$ is a ${\sigma}$-Stone space (a Stone space with the property that the closure of any countable union of clopen sets is clopen), ${{\mathcal X}}$ is the Baire ${\sigma}$-algebra (generated by the clopen sets), and the null sets are the meager sets. However, we will not use this additional structure here.

The partial inversion relationship between the functors ${F: \textbf{Prb} \rightarrow \textbf{AlgPrb}}$ and ${G: \textbf{AlgPrb} \rightarrow \textbf{Prb}}$ is given by the following assertion:

1. There is a natural transformation from ${F \circ G: \textbf{AlgPrb} \rightarrow \textbf{AlgPrb}}$ to the identity functor ${I: \textbf{AlgPrb} \rightarrow \textbf{AlgPrb}}$.

More informally: if one starts with an algebraic probability space ${({\mathcal A},\tau)}$ and converts it back into a classical probability space ${(X, {\mathcal X}, \mu)}$, then there is a trace-preserving algebra homomorphism of ${{\mathcal A}}$ to ${L^\infty( X, {\mathcal X}, \mu )}$, which respects morphisms of the algebraic probability space. While this relationship is far weaker than an equivalence of categories (which would require that ${F \circ G}$ and ${G \circ F}$ are both natural isomorphisms), it is still good enough to allow many ergodic theory problems formulated using classical probability spaces to be reformulated instead as an equivalent problem in algebraic probability spaces.

Remark 2 The opposite composition ${G \circ F: \textbf{Prb} \rightarrow \textbf{Prb}}$ is a little odd: it takes an arbitrary probability space ${(X, {\mathcal X}, \mu)}$ and returns a more complicated probability space ${(X', {\mathcal X}', \mu')}$, with ${X'}$ being the space of homomorphisms ${\iota_x: L^\infty(X, {\mathcal X}, \mu) \rightarrow {\bf R}}$. while there is “morally” an embedding of ${X}$ into ${X'}$ using the evaluation map, this map does not exist in general because points in ${X}$ may well have zero measure. However, if one takes a “pointless” approach and focuses just on the measure algebras ${({\mathcal X}, \mu)}$, ${({\mathcal X}', \mu')}$, then these algebras become naturally isomorphic after quotienting out by null sets.

Remark 3 An algebraic probability space captures a bit more structure than a classical probability space, because ${{\mathcal A}}$ may be identified with a proper subset of ${L^\infty}$ that describes the “regular” functions (or random variables) of the space. For instance, starting with the unit circle ${{\bf R}/{\bf Z}}$ (with the usual Haar measure and the usual trace ${\tau(f) = \int_{{\bf R}/{\bf Z}} f}$), any unital subalgebra ${{\mathcal A}}$ of ${L^\infty({\bf R}/{\bf Z})}$ that is dense in ${L^2({\bf R}/{\bf Z})}$ will generate the same classical probability space ${G( {\mathcal A}, \tau )}$ on applying the functor ${G}$, namely one will get the space ${({\bf R}/{\bf Z})'}$ of homomorphisms from ${L^\infty({\bf R}/{\bf Z})}$ to ${{\bf R}}$ (with the measure induced from ${\tau}$). Thus for instance ${{\mathcal A}}$ could be the continuous functions ${C( {\bf R}/{\bf Z} )}$, the Wiener algebra ${A({\bf R}/{\bf Z})}$ or the full space ${L^\infty({\bf R}/{\bf Z})}$, but the classical space ${G( {\mathcal A}, \tau )}$ will be unable to distinguish these spaces from each other. In particular, the functor ${F \circ G}$ loses information (roughly speaking, this functor takes an algebraic probability space and completes it to a von Neumann algebra, but then forgets exactly what algebra was initially used to create this completion). In ergodic theory, this sort of “extra structure” is traditionally encoded in topological terms, by assuming that the underlying probability space ${X}$ has a nice topological structure (e.g. a standard Borel space); however, with the algebraic perspective one has the freedom to have non-topological notions of extra structure, by choosing ${{\mathcal A}}$ to be something other than an algebra ${C(X)}$ of continuous functions on a topological space. I hope to discuss one such example of extra structure (coming from the Gowers-Host-Kra theory of uniformity seminorms) in a later blog post (this generalises the example of the Wiener algebra given previously, which is encoding “Fourier structure”).

A small example of how one could use the functors ${F, G}$ is as follows. Suppose one has a classical probability space ${(X, {\mathcal X}, \mu)}$ with a measure-preserving action of an uncountable group ${\Gamma}$, which is only defined (and an action) up to almost everywhere equivalence; thus for instance for any set ${E}$ and any ${g, h \in \Gamma}$, ${T^{gh} E}$ and ${T^g T^h E}$ might not be exactly equal, but only equal up to a null set. For similar reasons, an element ${E}$ of the invariant factor ${{\mathcal X}^\Gamma}$ might not be exactly invariant with respect to ${\Gamma}$, but instead one only has ${T^g E}$ and ${E}$ equal up to null sets for each ${g \in \Gamma}$. One might like to “clean up” the action of ${\Gamma}$ to make it defined everywhere, and a genuine action everywhere, but this is not immediately achievable if ${\Gamma}$ is uncountable, since the union of all the null sets where something bad occurs may cease to be a null set. However, by applying the functor ${F}$, each shift ${T^g: X \rightarrow X}$ defines a morphism ${T^g: L^\infty(X, {\mathcal X}, \mu) \rightarrow L^\infty(X, {\mathcal X}, \mu)}$ on the associated algebraic probability space (i.e. the Koopman operator), and then applying ${G}$, we obtain a shift ${T^g: X' \rightarrow X'}$ on a new classical probability space ${(X', {\mathcal X}', \mu')}$ which now gives a genuine measure-preserving action of ${\Gamma}$, and which is equivalent to the original action from a measure algebra standpoint. The invariant factor ${{\mathcal X}^\Gamma}$ now consists of those sets in ${{\mathcal X}'}$ which are genuinely ${\Gamma}$-invariant, not just up to null sets. (Basically, the classical probability space ${(X', {\mathcal X}', \mu')}$ contains a Boolean algebra ${\overline{\mathcal B}}$ with the property that every measurable set ${A \in {\mathcal X}'}$ is equivalent up to null sets to precisely one set in ${\overline{\mathcal B}}$, allowing for a canonical “retraction” onto ${\overline{\mathcal B}}$ that eliminates all null set issues.)

More indirectly, the functors ${F, G}$ suggest that one should be able to develop a “pointless” form of ergodic theory, in which the underlying probability spaces are given algebraically rather than classically. I hope to give some more specific examples of this in later posts.

There are a number of ways to construct the real numbers ${{\bf R}}$, for instance

• as the metric completion of ${{\bf Q}}$ (thus, ${{\bf R}}$ is defined as the set of Cauchy sequences of rationals, modulo Cauchy equivalence);
• as the space of Dedekind cuts on the rationals ${{\bf Q}}$;
• as the space of quasimorphisms ${\phi: {\bf Z} \rightarrow {\bf Z}}$ on the integers, quotiented by bounded functions. (I believe this construction first appears in this paper of Street, who credits the idea to Schanuel, though the germ of this construction arguably goes all the way back to Eudoxus.)

There is also a fourth family of constructions that proceeds via nonstandard analysis, as a special case of what is known as the nonstandard hull construction. (Here I will assume some basic familiarity with nonstandard analysis and ultraproducts, as covered for instance in this previous blog post.) Given an unbounded nonstandard natural number ${N \in {}^* {\bf N} \backslash {\bf N}}$, one can define two external additive subgroups of the nonstandard integers ${{}^* {\bf Z}}$:

• The group ${O(N) := \{ n \in {}^* {\bf Z}: |n| \leq CN \hbox{ for some } C \in {\bf N} \}}$ of all nonstandard integers of magnitude less than or comparable to ${N}$; and
• The group ${o(N) := \{ n \in {}^* {\bf Z}: |n| \leq C^{-1} N \hbox{ for all } C \in {\bf N} \}}$ of nonstandard integers of magnitude infinitesimally smaller than ${N}$.

The group ${o(N)}$ is a subgroup of ${O(N)}$, so we may form the quotient group ${O(N)/o(N)}$. This space is isomorphic to the reals ${{\bf R}}$, and can in fact be used to construct the reals:

Proposition 1 For any coset ${n + o(N)}$ of ${O(N)/o(N)}$, there is a unique real number ${\hbox{st} \frac{n}{N}}$ with the property that ${\frac{n}{N} = \hbox{st} \frac{n}{N} + o(1)}$. The map ${n + o(N) \mapsto \hbox{st} \frac{n}{N}}$ is then an isomorphism between the additive groups ${O(N)/o(N)}$ and ${{\bf R}}$.

Proof: Uniqueness is clear. For existence, observe that the set ${\{ x \in {\bf R}: Nx \leq n + o(N) \}}$ is a Dedekind cut, and its supremum can be verified to have the required properties for ${\hbox{st} \frac{n}{N}}$. $\Box$

In a similar vein, we can view the unit interval ${[0,1]}$ in the reals as the quotient

$\displaystyle [0,1] \equiv [N] / o(N) \ \ \ \ \ (1)$

where ${[N]}$ is the nonstandard (i.e. internal) set ${\{ n \in {\bf N}: n \leq N \}}$; of course, ${[N]}$ is not a group, so one should interpret ${[N]/o(N)}$ as the image of ${[N]}$ under the quotient map ${{}^* {\bf Z} \rightarrow {}^* {\bf Z} / o(N)}$ (or ${O(N) \rightarrow O(N)/o(N)}$, if one prefers). Or to put it another way, (1) asserts that ${[0,1]}$ is the image of ${[N]}$ with respect to the map ${\pi: n \mapsto \hbox{st} \frac{n}{N}}$.

In this post I would like to record a nice measure-theoretic version of the equivalence (1), which essentially appears already in standard texts on Loeb measure (see e.g. this text of Cutland). To describe the results, we must first quickly recall the construction of Loeb measure on ${[N]}$. Given an internal subset ${A}$ of ${[N]}$, we may define the elementary measure ${\mu_0(A)}$ of ${A}$ by the formula

$\displaystyle \mu_0(A) := \hbox{st} \frac{|A|}{N}.$

This is a finitely additive probability measure on the Boolean algebra of internal subsets of ${[N]}$. We can then construct the Loeb outer measure ${\mu^*(A)}$ of any subset ${A \subset [N]}$ in complete analogy with Lebesgue outer measure by the formula

$\displaystyle \mu^*(A) := \inf \sum_{n=1}^\infty \mu_0(A_n)$

where ${(A_n)_{n=1}^\infty}$ ranges over all sequences of internal subsets of ${[N]}$ that cover ${A}$. We say that a subset ${A}$ of ${[N]}$ is Loeb measurable if, for any (standard) ${\epsilon>0}$, one can find an internal subset ${B}$ of ${[N]}$ which differs from ${A}$ by a set of Loeb outer measure at most ${\epsilon}$, and in that case we define the Loeb measure ${\mu(A)}$ of ${A}$ to be ${\mu^*(A)}$. It is a routine matter to show (e.g. using the Carathéodory extension theorem) that the space ${{\mathcal L}}$ of Loeb measurable sets is a ${\sigma}$-algebra, and that ${\mu}$ is a countably additive probability measure on this space that extends the elementary measure ${\mu_0}$. Thus ${[N]}$ now has the structure of a probability space ${([N], {\mathcal L}, \mu)}$.

Now, the group ${o(N)}$ acts (Loeb-almost everywhere) on the probability space ${[N]}$ by the addition map, thus ${T^h n := n+h}$ for ${n \in [N]}$ and ${h \in o(N)}$ (excluding a set of Loeb measure zero where ${n+h}$ exits ${[N]}$). This action is clearly seen to be measure-preserving. As such, we can form the invariant factor ${Z^0_{o(N)}([N]) = ([N], {\mathcal L}^{o(N)}, \mu\downharpoonright_{{\mathcal L}^{o(N)}})}$, defined by restricting attention to those Loeb measurable sets ${A \subset [N]}$ with the property that ${T^h A}$ is equal ${\mu}$-almost everywhere to ${A}$ for each ${h \in o(N)}$.

The claim is then that this invariant factor is equivalent (up to almost everywhere equivalence) to the unit interval ${[0,1]}$ with Lebesgue measure ${m}$ (and the trivial action of ${o(N)}$), by the same factor map ${\pi: n \mapsto \hbox{st} \frac{n}{N}}$ used in (1). More precisely:

Theorem 2 Given a set ${A \in {\mathcal L}^{o(N)}}$, there exists a Lebesgue measurable set ${B \subset [0,1]}$, unique up to ${m}$-a.e. equivalence, such that ${A}$ is ${\mu}$-a.e. equivalent to the set ${\pi^{-1}(B) := \{ n \in [N]: \hbox{st} \frac{n}{N} \in B \}}$. Conversely, if ${B \in [0,1]}$ is Lebesgue measurable, then ${\pi^{-1}(B)}$ is in ${{\mathcal L}^{o(N)}}$, and ${\mu( \pi^{-1}(B) ) = m( B )}$.

$\displaystyle [0,1] \equiv Z^0_{o(N)}( [N] )$

of (1).

Proof: We first prove the converse. It is clear that ${\pi^{-1}(B)}$ is ${o(N)}$-invariant, so it suffices to show that ${\pi^{-1}(B)}$ is Loeb measurable with Loeb measure ${m(B)}$. This is easily verified when ${B}$ is an elementary set (a finite union of intervals). By countable subadditivity of outer measure, this implies that Loeb outer measure of ${\pi^{-1}(E)}$ is bounded by the Lebesgue outer measure of ${E}$ for any set ${E \subset [0,1]}$; since every Lebesgue measurable set differs from an elementary set by a set of arbitrarily small Lebesgue outer measure, the claim follows.

Now we establish the forward claim. Uniqueness is clear from the converse claim, so it suffices to show existence. Let ${A \in {\mathcal L}^{o(N)}}$. Let ${\epsilon>0}$ be an arbitrary standard real number, then we can find an internal set ${A_\epsilon \subset [N]}$ which differs from ${A}$ by a set of Loeb measure at most ${\epsilon}$. As ${A}$ is ${o(N)}$-invariant, we conclude that for every ${h \in o(N)}$, ${A_\epsilon}$ and ${T^h A_\epsilon}$ differ by a set of Loeb measure (and hence elementary measure) at most ${2\epsilon}$. By the (contrapositive of the) underspill principle, there must exist a standard ${\delta>0}$ such that ${A_\epsilon}$ and ${T^h A_\epsilon}$ differ by a set of elementary measure at most ${2\epsilon}$ for all ${|h| \leq \delta N}$. If we then define the nonstandard function ${f_\epsilon: [N] \rightarrow {}^* {\bf R}}$ by the formula

$\displaystyle f(n) := \hbox{st} \frac{1}{\delta N} \sum_{m \in [N]: m \leq n \leq m+\delta N} 1_{A_\epsilon}(m),$

then from the (nonstandard) triangle inequality we have

$\displaystyle \frac{1}{N} \sum_{n \in [N]} |f(n) - 1_{A_\epsilon}(n)| \leq 3\epsilon$

(say). On the other hand, ${f}$ has the Lipschitz continuity property

$\displaystyle |f(n)-f(m)| \leq \frac{2|n-m|}{\delta N}$

and so in particular we see that

$\displaystyle \hbox{st} f(n) = \tilde f( \hbox{st} \frac{n}{N} )$

for some Lipschitz continuous function ${\tilde f: [0,1] \rightarrow [0,1]}$. If we then let ${E_\epsilon}$ be the set where ${\tilde f \geq 1 - \sqrt{\epsilon}}$, one can check that ${A_\epsilon}$ differs from ${\pi^{-1}(E_\epsilon)}$ by a set of Loeb outer measure ${O(\sqrt{\epsilon})}$, and hence ${A}$ does so also. Sending ${\epsilon}$ to zero, we see (from the converse claim) that ${1_{E_\epsilon}}$ is a Cauchy sequence in ${L^1}$ and thus converges in ${L^1}$ for some Lebesgue measurable ${E}$. The sets ${A_\epsilon}$ then converge in Loeb outer measure to ${\pi^{-1}(E)}$, giving the claim. $\Box$

Thanks to the Lebesgue differentiation theorem, the conditional expectation ${{\bf E}( f | Z^0_{o(N)}([N]))}$ of a bounded Loeb-measurable function ${f: [N] \rightarrow {\bf R}}$ can be expressed (as a function on ${[0,1]}$, defined ${m}$-a.e.) as

$\displaystyle {\bf E}( f | Z^0_{o(N)}([N]))(x) := \lim_{\epsilon \rightarrow 0} \frac{1}{2\epsilon} \int_{[x-\epsilon N,x+\epsilon N]} f\ d\mu.$

By the abstract ergodic theorem from the previous post, one can also view this conditional expectation as the element in the closed convex hull of the shifts ${T^h f}$, ${h = o(N)}$ of minimal ${L^2}$ norm. In particular, we obtain a form of the von Neumann ergodic theorem in this context: the averages ${\frac{1}{H} \sum_{h=1}^H T^h f}$ for ${H=O(N)}$ converge (as a net, rather than a sequence) in ${L^2}$ to ${{\bf E}( f | Z^0_{o(N)}([N]))}$.

If ${f: [N] \rightarrow [-1,1]}$ is (the standard part of) an internal function, that is to say the ultralimit of a sequence ${f_n: [N_n] \rightarrow [-1,1]}$ of finitary bounded functions, one can view the measurable function ${F := {\bf E}( f | Z^0_{o(N)}([N]))}$ as a limit of the ${f_n}$ that is analogous to the “graphons” that emerge as limits of graphs (see e.g. the recent text of Lovasz on graph limits). Indeed, the measurable function ${F: [0,1] \rightarrow [-1,1]}$ is related to the discrete functions ${f_n: [N_n] \rightarrow [-1,1]}$ by the formula

$\displaystyle \int_a^b F(x)\ dx = \hbox{st} \lim_{n \rightarrow p} \frac{1}{N_n} \sum_{a N_n \leq m \leq b N_n} f_n(m)$

for all ${0 \leq a < b \leq 1}$, where ${p}$ is the nonprincipal ultrafilter used to define the nonstandard universe. In particular, from the Arzela-Ascoli diagonalisation argument there is a subsequence ${n_j}$ such that

$\displaystyle \int_a^b F(x)\ dx = \lim_{j \rightarrow \infty} \frac{1}{N_{n_j}} \sum_{a N_{n_j} \leq m \leq b N_{n_j}} f_n(m),$

thus ${F}$ is the asymptotic density function of the ${f_n}$. For instance, if ${f_n}$ is the indicator function of a randomly chosen subset of ${[N_n]}$, then the asymptotic density function would equal ${1/2}$ (almost everywhere, at least).

I’m continuing to look into understanding the ergodic theory of ${o(N)}$ actions, as I believe this may allow one to apply ergodic theory methods to the “single-scale” or “non-asymptotic” setting (in which one averages only over scales comparable to a large parameter ${N}$, rather than the traditional asymptotic approach of letting the scale go to infinity). I’m planning some further posts in this direction, though this is still a work in progress.

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if ${U: H \rightarrow H}$ is a unitary operator on a Hilbert space ${H}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \pi_{H^U} v$

in the strong topology, where ${H^U := \{ w \in H: Uw = w \}}$ is the ${U}$-invariant subspace of ${H}$, and ${\pi_{H^U}}$ is the orthogonal projection to ${H^U}$. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if ${G}$ is a countable amenable group acting on a Hilbert space ${H}$ by unitary transformations ${g: H \rightarrow H}$, and ${v \in H}$ is a vector in that Hilbert space, then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv = \pi_{H^G} v \ \ \ \ \ (1)$

for any Folner sequence ${\Phi_N}$ of ${G}$, where ${H^G := \{ w \in H: gw = w \hbox{ for all }g \in G \}}$ is the ${G}$-invariant subspace. Thus one can interpret ${\pi_{H^G} v}$ as a certain average of elements of the orbit ${Gv := \{ gv: g \in G \}}$ of ${v}$.

I recently discovered that there is a simple variant of this ergodic theorem that holds even when the group ${G}$ is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem) Let ${G}$ be an arbitrary group acting unitarily on a Hilbert space ${H}$, and let ${v}$ be a vector in ${H}$. Then ${\pi_{H^G} v}$ is the element in the closed convex hull of ${Gv := \{ gv: g \in G \}}$ of minimal norm, and is also the unique element of ${H^G}$ in this closed convex hull.

Proof: As the closed convex hull of ${Gv}$ is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element ${F}$ of minimal norm. If ${T_g F \neq F}$ for some ${g}$, then the midpoint of ${T_g F}$ and ${F}$ would be in the closed convex hull and be of smaller norm, a contradiction; thus ${F}$ is ${G}$-invariant. To finish the first claim, it suffices to show that ${v-F}$ is orthogonal to every element ${h}$ of ${H^G}$. But if this were not the case for some such ${h}$, we would have ${\langle T_g v - F, h \rangle = \langle v-F,h\rangle \neq 0}$ for all ${g \in G}$, and thus on taking convex hulls ${\langle F-F,h\rangle = \langle f-F,f\rangle \neq 0}$, a contradiction.

Finally, since ${T_g v - F}$ is orthogonal to ${H^G}$, the same is true for ${F'-F}$ for any ${F'}$ in the closed convex hull of ${Gv}$, and this gives the second claim. $\Box$

This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any ${\epsilon>0}$, Theorem 1 implies that there is a finite convex combination ${v_\epsilon}$ of shifts ${gv}$ of ${v}$ which lies within ${\epsilon}$ (in the ${H}$ norm) to ${\pi_{H^G} v}$. By the triangle inequality, all the averages ${\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv_\epsilon}$ also lie within ${\epsilon}$ of ${\pi_{H^G} v}$, but by the Folner property this implies that the averages ${\frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} gv}$ are eventually within ${2\epsilon}$ (say) of ${\pi_{H^G} v}$, giving the claim.

It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:

Corollary 2 (Relative orthogonality) Let ${G}$ be a group acting unitarily on a Hilbert space ${H}$, and let ${V}$ be a ${G}$-invariant subspace of ${H}$. Then ${V}$ and ${H^G}$ are relatively orthogonal over their common subspace ${V^G}$, that is to say the restrictions of ${V}$ and ${H^G}$ to the orthogonal complement of ${V^G}$ are orthogonal to each other.

Proof: By Theorem 1, we have ${\pi_{H^G} v = \pi_{V^G} v}$ for all ${v \in V}$, and the claim follows. (Thanks to Gergely Harcos for this short argument.) $\Box$

Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups ${G}$. Define a ${G}$-system ${(X, {\mathcal X}, \mu, (T_g)_{g \in G})}$ to be a probability space ${X = (X, {\mathcal X}, \mu)}$ together with a measure-preserving action ${(T_g)_{g \in G}}$ of ${G}$ on ${X}$; this gives an action of ${G}$ on ${L^2(X) = L^2(X,{\mathcal X},\mu)}$, which by abuse of notation we also call ${T_g}$:

$\displaystyle T_g f := f \circ T_{g^{-1}}.$

(In this post we follow the usual convention of defining the ${L^p}$ spaces by quotienting out by almost everywhere equivalence.) We say that a ${G}$-system is ergodic if ${L^2(X)^G}$ consists only of the constants.

(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space ${X}$ and quotienting ${{\mathcal X}}$ by the ${\sigma}$-ideal of null sets, and considering maps such as ${T_g}$ only on this quotient ${\sigma}$-algebra (or on the associated von Neumann algebra ${L^\infty(X)}$ or Hilbert space ${L^2(X)}$). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)

A factor ${Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}$ of a ${G}$-system ${X = (X,{\mathcal X},\mu, (T_g)_{g \in G})}$ is another ${G}$-system together with a factor map ${\pi: X \rightarrow Y}$ which commutes with the ${G}$-action (thus ${T_g \pi = \pi S_g}$ for all ${g \in G}$) and respects the measure in the sense that ${\mu(\pi^{-1}(E)) = \nu(E)}$ for all ${E \in {\mathcal Y}}$. For instance, the ${G}$-invariant factor ${Z^0_G(X) := (X, {\mathcal X}^G, \mu\downharpoonright_{{\mathcal X}^G}, (T_g)_{g \in G})}$, formed by restricting ${X}$ to the invariant algebra ${{\mathcal X}^G := \{ E \in {\mathcal X}: T_g E = E \hbox{ a.e. for all } g \in G \}}$, is a factor of ${X}$. (This factor is the first factor in an important hierachy, the next element of which is the Kronecker factor ${Z^1_G(X)}$, but we will not discuss higher elements of this hierarchy further here.) If ${Y}$ is a factor of ${X}$, we refer to ${X}$ as an extension of ${Y}$.

From Corollary 2 we have

Corollary 3 (Relative independence) Let ${X}$ be a ${G}$-system for a group ${G}$, and let ${Y}$ be a factor of ${X}$. Then ${Y}$ and ${Z^0_G(X)}$ are relatively independent over their common factor ${Z^0_G(Y)}$, in the sense that the spaces ${L^2(Y)}$ and ${L^2(Z^0_G(X))}$ are relatively orthogonal over ${L^2(Z^0_G(Y))}$ when all these spaces are embedded into ${L^2(X)}$.

This has a simple consequence regarding the product ${X \times Y = (X \times Y, {\mathcal X} \times {\mathcal Y}, \mu \times \nu, (T_g \oplus S_g)_{g \in G})}$ of two ${G}$-systems ${X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})}$ and ${Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}$, in the case when the ${Y}$ action is trivial:

Lemma 4 If ${X,Y}$ are two ${G}$-systems, with the action of ${G}$ on ${Y}$ trivial, then ${Z^0_G(X \times Y)}$ is isomorphic to ${Z^0_G(X) \times Y}$ in the obvious fashion.

This lemma is immediate for countable ${G}$, since for a ${G}$-invariant function ${f}$, one can ensure that ${T_g f = f}$ holds simultaneously for all ${g \in G}$ outside of a null set, but is a little trickier for uncountable ${G}$.

Proof: It is clear that ${Z^0_G(X) \times Y}$ is a factor of ${Z^0_G(X \times Y)}$. To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero ${f \in L^2(Z^0_G(X \times Y))}$ which is orthogonal to ${L^2(Z^0_G(X) \times Y)}$. In particular, we have ${fg}$ orthogonal to ${L^2(Z^0_G(X))}$ for any ${g \in L^\infty(Y)}$. Since ${fg}$ lies in ${L^2(Z^0_G(X \times Y))}$, we conclude from Corollary 3 (viewing ${X}$ as a factor of ${X \times Y}$) that ${fg}$ is also orthogonal to ${L^2(X)}$. Since ${g}$ is an arbitrary element of ${L^\infty(Y)}$, we conclude that ${f}$ is orthogonal to ${L^2(X \times Y)}$ and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.) $\Box$

Now we discuss the notion of a group extension.

Definition 5 (Group extension) Let ${G}$ be an arbitrary group, let ${Y = (Y, {\mathcal Y}, \nu, (S_g)_{g \in G})}$ be a ${G}$-system, and let ${K}$ be a compact metrisable group. A ${K}$-extension of ${Y}$ is an extension ${X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})}$ whose underlying space is ${X = Y \times K}$ (with ${{\mathcal X}}$ the product of ${{\mathcal Y}}$ and the Borel ${\sigma}$-algebra on ${K}$), the factor map is ${\pi: (y,k) \mapsto y}$, and the shift maps ${T_g}$ are given by

$\displaystyle T_g ( y, k ) = (S_g y, \rho_g(y) k )$

where for each ${g \in G}$, ${\rho_g: Y \rightarrow K}$ is a measurable map (known as the cocycle associated to the ${K}$-extension ${X}$).

An important special case of a ${K}$-extension arises when the measure ${\mu}$ is the product of ${\nu}$ with the Haar measure ${dk}$ on ${K}$. In this case, ${X}$ also has a ${K}$-action ${k': (y,k) \mapsto (y,k(k')^{-1})}$ that commutes with the ${G}$-action, making ${X}$ a ${G \times K}$-system. More generally, ${\mu}$ could be the product of ${\nu}$ with the Haar measure ${dh}$ of some closed subgroup ${H}$ of ${K}$, with ${\rho_g}$ taking values in ${H}$; then ${X}$ is now a ${G \times H}$ system. In this latter case we will call ${X}$ ${H}$-uniform.

If ${X}$ is a ${K}$-extension of ${Y}$ and ${U: Y \rightarrow K}$ is a measurable map, we can define the gauge transform ${X_U}$ of ${X}$ to be the ${K}$-extension of ${Y}$ whose measure ${\mu_U}$ is the pushforward of ${\mu}$ under the map ${(y,k) \mapsto (y, U(y) k)}$, and whose cocycles ${\rho_{g,U}: Y \rightarrow K}$ are given by the formula

$\displaystyle \rho_{g,U}(y) := U(gy) \rho_g(y) U(y)^{-1}.$

It is easy to see that ${X_U}$ is a ${K}$-extension that is isomorphic to ${X}$ as a ${K}$-extension of ${Y}$; we will refer to ${X_U}$ and ${X}$ as equivalent systems, and ${\rho_{g,U}}$ as cohomologous to ${\rho_g}$. We then have the following fundamental result of Mackey and of Zimmer:

Theorem 6 (Mackey-Zimmer theorem) Let ${G}$ be an arbitrary group, let ${Y}$ be an ergodic ${G}$-system, and let ${K}$ be a compact metrisable group. Then every ergodic ${K}$-extension ${X}$ of ${Y}$ is equivalent to an ${H}$-uniform extension of ${Y}$ for some closed subgroup ${H}$ of ${K}$.

This theorem is usually stated for amenable groups ${G}$, but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem, ${X}$ and ${Y}$ are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions via the ergodic decomposition, but the result becomes more complicated to state, and we will not do so here.

Given two unit vectors ${v,w}$ in a real inner product space, one can define the correlation between these vectors to be their inner product ${\langle v, w \rangle}$, or in more geometric terms, the cosine of the angle ${\angle(v,w)}$ subtended by ${v}$ and ${w}$. By the Cauchy-Schwarz inequality, this is a quantity between ${-1}$ and ${+1}$, with the extreme positive correlation ${+1}$ occurring when ${v,w}$ are identical, the extreme negative correlation ${-1}$ occurring when ${v,w}$ are diametrically opposite, and the zero correlation ${0}$ occurring when ${v,w}$ are orthogonal. This notion is closely related to the notion of correlation between two non-constant square-integrable real-valued random variables ${X,Y}$, which is the same as the correlation between two unit vectors ${v,w}$ lying in the Hilbert space ${L^2(\Omega)}$ of square-integrable random variables, with ${v}$ being the normalisation of ${X}$ defined by subtracting off the mean ${\mathbf{E} X}$ and then dividing by the standard deviation of ${X}$, and similarly for ${w}$ and ${Y}$.

One can also define correlation for complex (Hermitian) inner product spaces by taking the real part ${\hbox{Re} \langle , \rangle}$ of the complex inner product to recover a real inner product.

While reading the (highly recommended) recent popular maths book “How not to be wrong“, by my friend and co-author Jordan Ellenberg, I came across the (important) point that correlation is not necessarily transitive: if ${X}$ correlates with ${Y}$, and ${Y}$ correlates with ${Z}$, then this does not imply that ${X}$ correlates with ${Z}$. A simple geometric example is provided by the three unit vectors

$\displaystyle u := (1,0); v := (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}); w := (0,1)$

in the Euclidean plane ${{\bf R}^2}$: ${u}$ and ${v}$ have a positive correlation of ${\frac{1}{\sqrt{2}}}$, as does ${v}$ and ${w}$, but ${u}$ and ${w}$ are not correlated with each other. Or: for a typical undergraduate course, it is generally true that good exam scores are correlated with a deep understanding of the course material, and memorising from flash cards are correlated with good exam scores, but this does not imply that memorising flash cards is correlated with deep understanding of the course material.

However, there are at least two situations in which some partial version of transitivity of correlation can be recovered. The first is in the “99%” regime in which the correlations are very close to ${1}$: if ${u,v,w}$ are unit vectors such that ${u}$ is very highly correlated with ${v}$, and ${v}$ is very highly correlated with ${w}$, then this does imply that ${u}$ is very highly correlated with ${w}$. Indeed, from the identity

$\displaystyle \| u-v \| = 2^{1/2} (1 - \langle u,v\rangle)^{1/2}$

(and similarly for ${v-w}$ and ${u-w}$) and the triangle inequality

$\displaystyle \|u-w\| \leq \|u-v\| + \|v-w\|,$

we see that

$\displaystyle (1 - \langle u,w \rangle)^{1/2} \leq (1 - \langle u,v\rangle)^{1/2} + (1 - \langle v,w\rangle)^{1/2}. \ \ \ \ \ (1)$

Thus, for instance, if ${\langle u, v \rangle \geq 1-\varepsilon}$ and ${\langle v,w \rangle \geq 1-\varepsilon}$, then ${\langle u,w \rangle \geq 1-4\varepsilon}$. This is of course closely related to (though slightly weaker than) the triangle inequality for angles:

$\displaystyle \angle(u,w) \leq \angle(u,v) + \angle(v,w).$

Remark 1 (Thanks to Andrew Granville for conversations leading to this observation.) The inequality (1) also holds for sub-unit vectors, i.e. vectors ${u,v,w}$ with ${\|u\|, \|v\|, \|w\| \leq 1}$. This comes by extending ${u,v,w}$ in directions orthogonal to all three original vectors and to each other in order to make them unit vectors, enlarging the ambient Hilbert space ${H}$ if necessary. More concretely, one can apply (1) to the unit vectors

$\displaystyle (u, \sqrt{1-\|u\|^2}, 0, 0), (v, 0, \sqrt{1-\|v\|^2}, 0), (w, 0, 0, \sqrt{1-\|w\|^2})$

in ${H \times {\bf R}^3}$.

But even in the “${1\%}$” regime in which correlations are very weak, there is still a version of transitivity of correlation, known as the van der Corput lemma, which basically asserts that if a unit vector ${v}$ is correlated with many unit vectors ${u_1,\dots,u_n}$, then many of the pairs ${u_i,u_j}$ will then be correlated with each other. Indeed, from the Cauchy-Schwarz inequality

$\displaystyle |\langle v, \sum_{i=1}^n u_i \rangle|^2 \leq \|v\|^2 \| \sum_{i=1}^n u_i \|^2$

we see that

$\displaystyle (\sum_{i=1}^n \langle v, u_i \rangle)^2 \leq \sum_{1 \leq i,j \leq n} \langle u_i, u_j \rangle. \ \ \ \ \ (2)$

Thus, for instance, if ${\langle v, u_i \rangle \geq \varepsilon}$ for at least ${\varepsilon n}$ values of ${i=1,\dots,n}$, then (after removing those indices ${i}$ for which ${\langle v, u_i \rangle < \varepsilon}$) ${\sum_{i,j} \langle u_i, u_j \rangle}$ must be at least ${\varepsilon^4 n^2}$, which implies that ${\langle u_i, u_j \rangle \geq \varepsilon^4/2}$ for at least ${\varepsilon^4 n^2/2}$ pairs ${(i,j)}$. Or as another example: if a random variable ${X}$ exhibits at least ${1\%}$ positive correlation with ${n}$ other random variables ${Y_1,\dots,Y_n}$, then if ${n > 10,000}$, at least two distinct ${Y_i,Y_j}$ must have positive correlation with each other (although this argument does not tell you which pair ${Y_i,Y_j}$ are so correlated). Thus one can view this inequality as a sort of `pigeonhole principle” for correlation.

A similar argument (multiplying each ${u_i}$ by an appropriate sign ${\pm 1}$) shows the related van der Corput inequality

$\displaystyle (\sum_{i=1}^n |\langle v, u_i \rangle|)^2 \leq \sum_{1 \leq i,j \leq n} |\langle u_i, u_j \rangle|, \ \ \ \ \ (3)$

and this inequality is also true for complex inner product spaces. (Also, the ${u_i}$ do not need to be unit vectors for this inequality to hold.)

Geometrically, the picture is this: if ${v}$ positively correlates with all of the ${u_1,\dots,u_n}$, then the ${u_1,\dots,u_n}$ are all squashed into a somewhat narrow cone centred at ${v}$. The cone is still wide enough to allow a few pairs ${u_i, u_j}$ to be orthogonal (or even negatively correlated) with each other, but (when ${n}$ is large enough) it is not wide enough to allow all of the ${u_i,u_j}$ to be so widely separated. Remarkably, the bound here does not depend on the dimension of the ambient inner product space; while increasing the number of dimensions should in principle add more “room” to the cone, this effect is counteracted by the fact that in high dimensions, almost all pairs of vectors are close to orthogonal, and the exceptional pairs that are even weakly correlated to each other become exponentially rare. (See this previous blog post for some related discussion; in particular, Lemma 2 from that post is closely related to the van der Corput inequality presented here.)

A particularly common special case of the van der Corput inequality arises when ${v}$ is a unit vector fixed by some unitary operator ${T}$, and the ${u_i}$ are shifts ${u_i = T^i u}$ of a single unit vector ${u}$. In this case, the inner products ${\langle v, u_i \rangle}$ are all equal, and we arrive at the useful van der Corput inequality

$\displaystyle |\langle v, u \rangle|^2 \leq \frac{1}{n^2} \sum_{1 \leq i,j \leq n} |\langle T^i u, T^j u \rangle|. \ \ \ \ \ (4)$

(In fact, one can even remove the absolute values from the right-hand side, by using (2) instead of (4).) Thus, to show that ${v}$ has negligible correlation with ${u}$, it suffices to show that the shifts of ${u}$ have negligible correlation with each other.

Here is a basic application of the van der Corput inequality:

Proposition 2 (Weyl equidistribution estimate) Let ${P: {\bf Z} \rightarrow {\bf R}/{\bf Z}}$ be a polynomial with at least one non-constant coefficient irrational. Then one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N e( P(n) ) = 0,$

where ${e(x) := e^{2\pi i x}}$.

Note that this assertion implies the more general assertion

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N e( kP(n) ) = 0$

for any non-zero integer ${k}$ (simply by replacing ${P}$ by ${kP}$), which by the Weyl equidistribution criterion is equivalent to the sequence ${P(1), P(2),\dots}$ being asymptotically equidistributed in ${{\bf R}/{\bf Z}}$.

Proof: We induct on the degree ${d}$ of the polynomial ${P}$, which must be at least one. If ${d}$ is equal to one, the claim is easily established from the geometric series formula, so suppose that ${d>1}$ and that the claim has already been proven for ${d-1}$. If the top coefficient ${a_d}$ of ${P(n) = a_d n^d + \dots + a_0}$ is rational, say ${a_d = \frac{p}{q}}$, then by partitioning the natural numbers into residue classes modulo ${q}$, we see that the claim follows from the induction hypothesis; so we may assume that the top coefficient ${a_d}$ is irrational.

In order to use the van der Corput inequality as stated above (i.e. in the formalism of inner product spaces) we will need a non-principal ultrafilter ${p}$ (see e.g this previous blog post for basic theory of ultrafilters); we leave it as an exercise to the reader to figure out how to present the argument below without the use of ultrafilters (or similar devices, such as Banach limits). The ultrafilter ${p}$ defines an inner product ${\langle, \rangle_p}$ on bounded complex sequences ${z = (z_1,z_2,z_3,\dots)}$ by setting

$\displaystyle \langle z, w \rangle_p := \hbox{st} \lim_{N \rightarrow p} \frac{1}{N} \sum_{n=1}^N z_n \overline{w_n}.$

Strictly speaking, this inner product is only positive semi-definite rather than positive definite, but one can quotient out by the null vectors to obtain a positive-definite inner product. To establish the claim, it will suffice to show that

$\displaystyle \langle 1, e(P) \rangle_p = 0$

for every non-principal ultrafilter ${p}$.

Note that the space of bounded sequences (modulo null vectors) admits a shift ${T}$, defined by

$\displaystyle T (z_1,z_2,\dots) := (z_2,z_3,\dots).$

This shift becomes unitary once we quotient out by null vectors, and the constant sequence ${1}$ is clearly a unit vector that is invariant with respect to the shift. So by the van der Corput inequality, we have

$\displaystyle |\langle 1, e(P) \rangle_p| \leq \frac{1}{n^2} \sum_{1 \leq i,j \leq n} |\langle T^i e(P), T^j e(P) \rangle_p|$

for any ${n \geq 1}$. But we may rewrite ${\langle T^i e(P), T^j e(P) \rangle = \langle 1, e(T^j P - T^i P) \rangle_p}$. Then observe that if ${i \neq j}$, ${T^j P - T^i P}$ is a polynomial of degree ${d-1}$ whose ${d-1}$ coefficient is irrational, so by induction hypothesis we have ${\langle T^i e(P), T^j e(P) \rangle_p = 0}$ for ${i \neq j}$. For ${i=j}$ we of course have ${\langle T^i e(P), T^j e(P) \rangle_p = 1}$, and so

$\displaystyle |\langle 1, e(P) \rangle_p| \leq \frac{1}{n^2} \times n$

for any ${n}$. Letting ${n \rightarrow \infty}$, we obtain the claim. $\Box$