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I’ve just uploaded to the arXiv my paper “The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture“. As the title suggests, this paper links together the Elliott-Halberstam conjecture from sieve theory with the conjecture of Vinogradov concerning the least quadratic nonresidue {n(p)} of a prime {p}. Vinogradov established the bound

\displaystyle  n(p) \ll p^{\frac{1}{2\sqrt{e}}} \log^2 p \ \ \ \ \ (1)

and conjectured that

\displaystyle  n(p) \ll p^\varepsilon \ \ \ \ \ (2)

for any fixed {\varepsilon>0}. Unconditionally, the best result so far (up to logarithmic factors) that holds for all primes {p} is by Burgess, who showed that

\displaystyle  n(p) \ll p^{\frac{1}{4\sqrt{e}}+\varepsilon} \ \ \ \ \ (3)

for any fixed {\varepsilon>0}. See this previous post for a proof of these bounds.

In this paper, we show that the Vinogradov conjecture is a consequence of the Elliott-Halberstam conjecture. Using a variant of the argument, we also show that the “Type II” estimates established by Zhang and numerically improved by the Polymath8a project can be used to improve a little on the Vinogradov bound (1), to

\displaystyle  n(p) \ll p^{(\frac{1}{2}-\frac{1}{34})\frac{1}{\sqrt{e}} + \varepsilon},

although this falls well short of the Burgess bound. However, the method is somewhat different (although in both cases it is the Weil exponential sum bounds that are the source of the gain over (1)) and it is conceivable that a numerically stronger version of the Type II estimates could obtain results that are more competitive with the Burgess bound. At any rate, this demonstrates that the equidistribution estimates introduced by Zhang may have further applications beyond the family of results related to bounded gaps between primes.

The connection between the least quadratic nonresidue problem and Elliott-Halberstam is follows. Suppose for contradiction we can find a prime {q} with {n(q)} unusually large. Letting {\chi} be the quadratic character modulo {q}, this implies that the sums {\sum_{n \leq x} \chi(n)} are also unusually large for a significant range of {x} (e.g. {x < n(q)}), although the sum is also quite small for large {x} (e.g. {x > q}), due to the cancellation present in {\chi}. It turns out (by a sort of “uncertainty principle” for multiplicative functions, as per this paper of Granville and Soundararajan) that these facts force {\sum_{n\leq x} \chi(n) \Lambda(n)} to be unusually large in magnitude for some large {x} (with {q^C \leq x \leq q^{C'}} for two large absolute constants {C,C'}). By the periodicity of {\chi}, this means that

\displaystyle  \sum_{n\leq x} \chi(n) \Lambda(n+q)

must be unusually large also. However, because {n(q)} is large, one can factorise {\chi} as {f * 1} for a fairly sparsely supported function {f = \chi * \mu}. The Elliott-Halberstam conjecture, which controls the distribution of {\Lambda} in arithmetic progressions on the average can then be used to show that {\sum_{n \leq x} (f*1)(n) \Lambda(n+q)} is small, giving the required contradiction.

The implication involving Type II estimates is proven by a variant of the argument. If {n(q)} is large, then a character sum {\sum_{N\leq n \leq 2N} \chi(n)} is unusually large for a certain {N}. By multiplicativity of {\chi}, this shows that {\chi} correlates with {\chi * 1_{[N,2N]}}, and then by periodicity of {\chi}, this shows that {\chi(n)} correlates with {\chi * 1_{[N,2N]}(n+jq)} for various small {q}. By the Cauchy-Schwarz inequality (cf. this previous blog post), this implies that {\chi * 1_{[N,2N]}(n+jq)} correlates with {\chi * 1_{[N,2N]}(n+j'q)} for some distinct {j,j'}. But this can be ruled out by using Type II estimates.

I’ll record here a well-known observation concerning potential counterexamples to any improvement to the Burgess bound, that is to say an infinite sequence of primes {p} with {n(p) = p^{\frac{1}{4\sqrt{e}} + o(1)}}. Suppose we let {a(t)} be the asymptotic mean value of the quadratic character {\chi} at {p^t} and {b(t)} the mean value of {\chi \Lambda}; these quantities are defined precisely in my paper, but roughly speaking one can think of

\displaystyle  a(t) = \lim_{p \rightarrow \infty} \frac{1}{p^t} \sum_{n \leq p^t} \chi(n)

and

\displaystyle  b(t) = \lim_{p \rightarrow \infty} \frac{1}{p^t} \sum_{n \leq p^t} \chi(n) \Lambda(n).

Thanks to the basic Dirichlet convolution identity {\chi(n) \log(n) = \chi * \chi\Lambda(n)}, one can establish the Wirsing integral equation

\displaystyle  t a(t) = \int_0^t a(u) b(t-u)\ du

for all {t \geq 0}; see my paper for details (actually far sharper claims than this appear in previous work of Wirsing and Granville-Soundararajan). If we have an infinite sequence of counterexamples to any improvement to the Burgess bound, then we have

\displaystyle  a(t)=b(t) = 1 \hbox{ for } t < \frac{1}{4\sqrt{e}}

while from the Burgess exponential sum estimates we have

\displaystyle  a(t) = 0 \hbox{ for } t > \frac{1}{4}.

These two constraints, together with the Wirsing integral equation, end up determining {a} and {b} completely. It turns out that we must have

\displaystyle  b(t) = -1 \hbox{ for } \frac{1}{4\sqrt{e}} \leq t \leq \frac{1}{4}

and

\displaystyle  a(t) = 1 - 2 \log(4 \sqrt{e} t) \hbox{ for } \frac{1}{4\sqrt{e}} \leq t \leq \frac{1}{4}

and then for {t > \frac{1}{4}}, {b} evolves by the integral equation

\displaystyle  b(t) = \int_{1/4\sqrt{e}}^{1/4} b(t-u) \frac{2du}{u}.

For instance

\displaystyle  b(t) = 1 \hbox{ for } \frac{1}{4} < t \leq \frac{1}{2\sqrt{e}}

and then {b} oscillates in a somewhat strange fashion towards zero as {t \rightarrow \infty}. This very odd behaviour of {\sum_n \chi(n) \Lambda(n)} is surely impossible, but it seems remarkably hard to exclude it without invoking a strong hypothesis, such as GRH or the Elliott-Halberstam conjecture (or weaker versions thereof).

The prime number theorem can be expressed as the assertion

\displaystyle  \sum_{n \leq x} \Lambda(n) = x + o(x) \ \ \ \ \ (1)

as {x \rightarrow \infty}, where

\displaystyle  \Lambda(n) := \sum_{d|n} \mu(d) \log \frac{n}{d}

is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula

\displaystyle  \sum_{n \leq x} \Lambda_2(n) = 2 x \log x + O(x) \ \ \ \ \ (2)

where the second von Mangoldt function {\Lambda_2} is defined by the formula

\displaystyle  \Lambda_2(n) := \sum_{d|n} \mu(d) \log^2 \frac{n}{d} \ \ \ \ \ (3)

or equivalently

\displaystyle  \Lambda_2(n) = \Lambda(n) \log n + \sum_{d|n} \Lambda(d) \Lambda(\frac{n}{d}). \ \ \ \ \ (4)

(We are avoiding the use of the {*} symbol here to denote Dirichlet convolution, as we will need this symbol to denote ordinary convolution shortly.) For the convenience of the reader, we give a proof of the Selberg symmetry formula below the fold. Actually, for the purposes of proving the prime number theorem, the weaker estimate

\displaystyle  \sum_{n \leq x} \Lambda_2(n) = 2 x \log x + o(x \log x) \ \ \ \ \ (5)

suffices.

In this post I would like to record a somewhat “soft analysis” reformulation of the elementary proof of the prime number theorem in terms of Banach algebras, and specifically in Banach algebra structures on (completions of) the space {C_c({\bf R})} of compactly supported continuous functions {f: {\bf R} \rightarrow {\bf C}} equipped with the convolution operation

\displaystyle  f*g(t) := \int_{\bf R} f(u) g(t-u)\ du.

This soft argument does not easily give any quantitative decay rate in the prime number theorem, but by the same token it avoids many of the quantitative calculations in the traditional proofs of this theorem. Ultimately, the key “soft analysis” fact used is the spectral radius formula

\displaystyle  \lim_{n \rightarrow \infty} \|f^n\|^{1/n} = \sup_{\lambda \in \hat B} |\lambda(f)| \ \ \ \ \ (6)

for any element {f} of a unital commutative Banach algebra {B}, where {\hat B} is the space of characters (i.e., continuous unital algebra homomorphisms from {B} to {{\bf C}}) of {B}. This formula is due to Gelfand and may be found in any text on Banach algebras; for sake of completeness we prove it below the fold.

The connection between prime numbers and Banach algebras is given by the following consequence of the Selberg symmetry formula.

Theorem 1 (Construction of a Banach algebra norm) For any {G \in C_c({\bf R})}, let {\|G\|} denote the quantity

\displaystyle  \|G\| := \limsup_{x \rightarrow \infty} |\sum_n \frac{\Lambda(n)}{n} G( \log \frac{x}{n} ) - \int_{\bf R} G(t)\ dt|.

Then {\| \|} is a seminorm on {C_c({\bf R})} with the bound

\displaystyle  \|G\| \leq \|G\|_{L^1({\bf R})} := \int_{\bf R} |G(t)|\ dt \ \ \ \ \ (7)

for all {G \in C_c({\bf R})}. Furthermore, we have the Banach algebra bound

\displaystyle  \| G * H \| \leq \|G\| \|H\| \ \ \ \ \ (8)

for all {G,H \in C_c({\bf R})}.

We prove this theorem below the fold. The prime number theorem then follows from Theorem 1 and the following two assertions. The first is an application of the spectral radius formula (6) and some basic Fourier analysis (in particular, the observation that {C_c({\bf R})} contains a plentiful supply of local units:

Theorem 2 (Non-trivial Banach algebras with many local units have non-trivial spectrum) Let {\| \|} be a seminorm on {C_c({\bf R})} obeying (7), (8). Suppose that {\| \|} is not identically zero. Then there exists {\xi \in {\bf R}} such that

\displaystyle  |\int_{\bf R} G(t) e^{-it\xi}\ dt| \leq \|G\|

for all {G \in C_c}. In particular, by (7), one has

\displaystyle  \|G\| = \| G \|_{L^1({\bf R})}

whenever {G(t) e^{-it\xi}} is a non-negative function.

The second is a consequence of the Selberg symmetry formula and the fact that {\Lambda} is real (as well as Mertens’ theorem, in the {\xi=0} case), and is closely related to the non-vanishing of the Riemann zeta function {\zeta} on the line {\{ 1+i\xi: \xi \in {\bf R}\}}:

Theorem 3 (Breaking the parity barrier) Let {\xi \in {\bf R}}. Then there exists {G \in C_c({\bf R})} such that {G(t) e^{-it\xi}} is non-negative, and

\displaystyle  \|G\| < \|G\|_{L^1({\bf R})}.

Assuming Theorems 1, 2, 3, we may now quickly establish the prime number theorem as follows. Theorem 2 and Theorem 3 imply that the seminorm {\| \|} constructed in Theorem 1 is trivial, and thus

\displaystyle  \sum_n \frac{\Lambda(n)}{n} G( \log \frac{x}{n} ) = \int_{\bf R} G(t)\ dt + o(1)

as {x \rightarrow \infty} for any Schwartz function {G} (the decay rate in {o(1)} may depend on {G}). Specialising to functions of the form {G(t) = e^{-t} \eta( e^{-t} )} for some smooth compactly supported {\eta} on {(0,+\infty)}, we conclude that

\displaystyle  \sum_n \Lambda(n) \eta(\frac{n}{x}) = \int_{\bf R} \eta(u)\ du + o(x)

as {x \rightarrow \infty}; by the smooth Urysohn lemma this implies that

\displaystyle  \sum_{\varepsilon x \leq n \leq x} \Lambda(n) = x - \varepsilon x + o(x)

as {x \rightarrow \infty} for any fixed {\varepsilon>0}, and the prime number theorem then follows by a telescoping series argument.

The same argument also yields the prime number theorem in arithmetic progressions, or equivalently that

\displaystyle  \sum_{n \leq x} \Lambda(n) \chi(n) = o(x)

for any fixed Dirichlet character {\chi}; the one difference is that the use of Mertens’ theorem is replaced by the basic fact that the quantity {L(1,\chi) = \sum_n \frac{\chi(n)}{n}} is non-vanishing.

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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.

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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}.

[UPDATE: Exercises corrected; thanks to Keith Conrad for some additional corrections and comments.]

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