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As readers who have followed my previous post will know, I have been spending the last few weeks extending my previous interactive text on propositional logic (entitied “QED”) to also cover first-order logic.  The text has now reached what seems to be a stable form, with a complete set of deductive rules for first-order logic with equality, and no major bugs as far as I can tell (apart from one weird visual bug I can’t eradicate, in that some graphics elements can occasionally temporarily disappear when one clicks on an item).  So it will likely not change much going forward.

I feel though that there could be more that could be done with this sort of framework (e.g., improved GUI, modification to other logics, developing the ability to write one’s own texts and libraries, exploring mathematical theories such as Peano arithmetic, etc.).  But writing this text (particularly the first-order logic sections) has brought me close to the limit of my programming ability, as the number of bugs introduced with each new feature implemented has begun to grow at an alarming rate.  I would like to repackage the code so that it can be re-used by more adept programmers for further possible applications, though I have never done something like this before and would appreciate advice on how to do so.   The code is already available under a Creative Commons licence, but I am not sure how readable and modifiable it will be to others currently.

[One thing I noticed is that I would probably have to make more of a decoupling between the GUI elements, the underlying logical elements, and the interactive text.  For instance, at some point I made the decision (convenient at the time) to use some GUI elements to store some of the state variables of the text, e.g. the exercise buttons are currently storing the status of what exercises are unlocked or not.  This is presumably not an example of good programming practice, though it would be relatively easy to fix.  More seriously, due to my inability to come up with a good general-purpose matching algorithm (or even specification of such an algorithm) for the the laws of first-order logic, many of the laws have to be hard-coded into the matching routine, so one cannot currently remove them from the text.  It may well be that the best thing to do in fact is to rework the entire codebase from scratch using more professional software design methods.]



About six years ago on this blog, I started thinking about trying to make a web-based game based around high-school algebra, and ended up using Scratch to write a short but playable puzzle game in which one solves linear equations for an unknown {x} using a restricted set of moves. (At almost the same time, there were a number of more professionally made games released along similar lines, most notably Dragonbox.)

Since then, I have thought a couple times about whether there were other parts of mathematics which could be gamified in a similar fashion. Shortly after my first blog posts on this topic, I experimented with a similar gamification of Lewis Carroll’s classic list of logic puzzles, but the results were quite clunky, and I was never satisfied with the results.

Over the last few weeks I returned to this topic though, thinking in particular about how to gamify the rules of inference of propositional logic, in a manner that at least vaguely resembles how mathematicians actually go about making logical arguments (e.g., splitting into cases, arguing by contradiction, using previous result as lemmas to help with subsequent ones, and so forth). The rules of inference are a list of a dozen or so deductive rules concerning propositional sentences (things like “({A} AND {B}) OR (NOT {C})”, where {A,B,C} are some formulas). A typical such rule is Modus Ponens: if the sentence {A} is known to be true, and the implication “{A} IMPLIES {B}” is also known to be true, then one can deduce that {B} is also true. Furthermore, in this deductive calculus it is possible to temporarily introduce some unproven statements as an assumption, only to discharge them later. In particular, we have the deduction theorem: if, after making an assumption {A}, one is able to derive the statement {B}, then one can conclude that the implication “{A} IMPLIES {B}” is true without any further assumption.

It took a while for me to come up with a workable game-like graphical interface for all of this, but I finally managed to set one up, now using Javascript instead of Scratch (which would be hopelessly inadequate for this task); indeed, part of the motivation of this project was to finally learn how to program in Javascript, which turned out to be not as formidable as I had feared (certainly having experience with other C-like languages like C++, Java, or lua, as well as some prior knowledge of HTML, was very helpful). The main code for this project is available here. Using this code, I have created an interactive textbook in the style of a computer game, which I have titled “QED”. This text contains thirty-odd exercises arranged in twelve sections that function as game “levels”, in which one has to use a given set of rules of inference, together with a given set of hypotheses, to reach a desired conclusion. The set of available rules increases as one advances through the text; in particular, each new section gives one or more rules, and additionally each exercise one solves automatically becomes a new deduction rule one can exploit in later levels, much as lemmas and propositions are used in actual mathematics to prove more difficult theorems. The text automatically tries to match available deduction rules to the sentences one clicks on or drags, to try to minimise the amount of manual input one needs to actually make a deduction.

Most of one’s proof activity takes place in a “root environment” of statements that are known to be true (under the given hypothesis), but for more advanced exercises one has to also work in sub-environments in which additional assumptions are made. I found the graphical metaphor of nested boxes to be useful to depict this tree of sub-environments, and it seems to combine well with the drag-and-drop interface.

The text also logs one’s moves in a more traditional proof format, which shows how the mechanics of the game correspond to a traditional mathematical argument. My hope is that this will give students a way to understand the underlying concept of forming a proof in a manner that is more difficult to achieve using traditional, non-interactive textbooks.

I have tried to organise the exercises in a game-like progression in which one first works with easy levels that train the player on a small number of moves, and then introduce more advanced moves one at a time. As such, the order in which the rules of inference are introduced is a little idiosyncratic. The most powerful rule (the law of the excluded middle, which is what separates classical logic from intuitionistic logic) is saved for the final section of the text.

Anyway, I am now satisfied enough with the state of the code and the interactive text that I am willing to make both available (and open source; I selected a CC-BY licence for both), and would be happy to receive feedback on any aspect of the either. In principle one could extend the game mechanics to other mathematical topics than the propositional calculus – the rules of inference for first-order logic being an obvious next candidate – but it seems to make sense to focus just on propositional logic for now.

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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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.

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    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 )}.

    More informally, we have the measure-theoretic version

    \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.

    (This is an extended blog post version of my talk “Ultraproducts as a Bridge Between Discrete and Continuous Analysis” that I gave at the Simons institute for the theory of computing at the workshop “Neo-Classical methods in discrete analysis“. Some of the material here is drawn from previous blog posts, notably “Ultraproducts as a bridge between hard analysis and soft analysis” and “Ultralimit analysis and quantitative algebraic geometry“‘. The text here has substantially more details than the talk; one may wish to skip all of the proofs given here to obtain a closer approximation to the original talk.)

    Discrete analysis, of course, is primarily interested in the study of discrete (or “finitary”) mathematical objects: integers, rational numbers (which can be viewed as ratios of integers), finite sets, finite graphs, finite or discrete metric spaces, and so forth. However, many powerful tools in mathematics (e.g. ergodic theory, measure theory, topological group theory, algebraic geometry, spectral theory, etc.) work best when applied to continuous (or “infinitary”) mathematical objects: real or complex numbers, manifolds, algebraic varieties, continuous topological or metric spaces, etc. In order to apply results and ideas from continuous mathematics to discrete settings, there are basically two approaches. One is to directly discretise the arguments used in continuous mathematics, which often requires one to keep careful track of all the bounds on various quantities of interest, particularly with regard to various error terms arising from discretisation which would otherwise have been negligible in the continuous setting. The other is to construct continuous objects as limits of sequences of discrete objects of interest, so that results from continuous mathematics may be applied (often as a “black box”) to the continuous limit, which then can be used to deduce consequences for the original discrete objects which are quantitative (though often ineffectively so). The latter approach is the focus of this current talk.

    The following table gives some examples of a discrete theory and its continuous counterpart, together with a limiting procedure that might be used to pass from the former to the latter:

    (Discrete) (Continuous) (Limit method)
    Ramsey theory Topological dynamics Compactness
    Density Ramsey theory Ergodic theory Furstenberg correspondence principle
    Graph/hypergraph regularity Measure theory Graph limits
    Polynomial regularity Linear algebra Ultralimits
    Structural decompositions Hilbert space geometry Ultralimits
    Fourier analysis Spectral theory Direct and inverse limits
    Quantitative algebraic geometry Algebraic geometry Schemes
    Discrete metric spaces Continuous metric spaces Gromov-Hausdorff limits
    Approximate group theory Topological group theory Model theory

    As the above table illustrates, there are a variety of different ways to form a limiting continuous object. Roughly speaking, one can divide limits into three categories:

    • Topological and metric limits. These notions of limits are commonly used by analysts. Here, one starts with a sequence (or perhaps a net) of objects {x_n} in a common space {X}, which one then endows with the structure of a topological space or a metric space, by defining a notion of distance between two points of the space, or a notion of open neighbourhoods or open sets in the space. Provided that the sequence or net is convergent, this produces a limit object {\lim_{n \rightarrow \infty} x_n}, which remains in the same space, and is “close” to many of the original objects {x_n} with respect to the given metric or topology.
    • Categorical limits. These notions of limits are commonly used by algebraists. Here, one starts with a sequence (or more generally, a diagram) of objects {x_n} in a category {X}, which are connected to each other by various morphisms. If the ambient category is well-behaved, one can then form the direct limit {\varinjlim x_n} or the inverse limit {\varprojlim x_n} of these objects, which is another object in the same category {X}, and is connected to the original objects {x_n} by various morphisms.
    • Logical limits. These notions of limits are commonly used by model theorists. Here, one starts with a sequence of objects {x_{\bf n}} or of spaces {X_{\bf n}}, each of which is (a component of) a model for given (first-order) mathematical language (e.g. if one is working in the language of groups, {X_{\bf n}} might be groups and {x_{\bf n}} might be elements of these groups). By using devices such as the ultraproduct construction, or the compactness theorem in logic, one can then create a new object {\lim_{{\bf n} \rightarrow \alpha} x_{\bf n}} or a new space {\prod_{{\bf n} \rightarrow \alpha} X_{\bf n}}, which is still a model of the same language (e.g. if the spaces {X_{\bf n}} were all groups, then the limiting space {\prod_{{\bf n} \rightarrow \alpha} X_{\bf n}} will also be a group), and is “close” to the original objects or spaces in the sense that any assertion (in the given language) that is true for the limiting object or space, will also be true for many of the original objects or spaces, and conversely. (For instance, if {\prod_{{\bf n} \rightarrow \alpha} X_{\bf n}} is an abelian group, then the {X_{\bf n}} will also be abelian groups for many {{\bf n}}.)

    The purpose of this talk is to highlight the third type of limit, and specifically the ultraproduct construction, as being a “universal” limiting procedure that can be used to replace most of the limits previously mentioned. Unlike the topological or metric limits, one does not need the original objects {x_{\bf n}} to all lie in a common space {X} in order to form an ultralimit {\lim_{{\bf n} \rightarrow \alpha} x_{\bf n}}; they are permitted to lie in different spaces {X_{\bf n}}; this is more natural in many discrete contexts, e.g. when considering graphs on {{\bf n}} vertices in the limit when {{\bf n}} goes to infinity. Also, no convergence properties on the {x_{\bf n}} are required in order for the ultralimit to exist. Similarly, ultraproduct limits differ from categorical limits in that no morphisms between the various spaces {X_{\bf n}} involved are required in order to construct the ultraproduct.

    With so few requirements on the objects {x_{\bf n}} or spaces {X_{\bf n}}, the ultraproduct construction is necessarily a very “soft” one. Nevertheless, the construction has two very useful properties which make it particularly useful for the purpose of extracting good continuous limit objects out of a sequence of discrete objects. First of all, there is Łos’s theorem, which roughly speaking asserts that any first-order sentence which is asymptotically obeyed by the {x_{\bf n}}, will be exactly obeyed by the limit object {\lim_{{\bf n} \rightarrow \alpha} x_{\bf n}}; in particular, one can often take a discrete sequence of “partial counterexamples” to some assertion, and produce a continuous “complete counterexample” that same assertion via an ultraproduct construction; taking the contrapositives, one can often then establish a rigorous equivalence between a quantitative discrete statement and its qualitative continuous counterpart. Secondly, there is the countable saturation property that ultraproducts automatically enjoy, which is a property closely analogous to that of compactness in topological spaces, and can often be used to ensure that the continuous objects produced by ultraproduct methods are “complete” or “compact” in various senses, which is particularly useful in being able to upgrade qualitative (or “pointwise”) bounds to quantitative (or “uniform”) bounds, more or less “for free”, thus reducing significantly the burden of “epsilon management” (although the price one pays for this is that one needs to pay attention to which mathematical objects of study are “standard” and which are “nonstandard”). To achieve this compactness or completeness, one sometimes has to restrict to the “bounded” portion of the ultraproduct, and it is often also convenient to quotient out the “infinitesimal” portion in order to complement these compactness properties with a matching “Hausdorff” property, thus creating familiar examples of continuous spaces, such as locally compact Hausdorff spaces.

    Ultraproducts are not the only logical limit in the model theorist’s toolbox, but they are one of the simplest to set up and use, and already suffice for many of the applications of logical limits outside of model theory. In this post, I will set out the basic theory of these ultraproducts, and illustrate how they can be used to pass between discrete and continuous theories in each of the examples listed in the above table.

    Apart from the initial “one-time cost” of setting up the ultraproduct machinery, the main loss one incurs when using ultraproduct methods is that it becomes very difficult to extract explicit quantitative bounds from results that are proven by transferring qualitative continuous results to the discrete setting via ultraproducts. However, in many cases (particularly those involving regularity-type lemmas) the bounds are already of tower-exponential type or worse, and there is arguably not much to be lost by abandoning the explicit quantitative bounds altogether.

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    The classical foundations of probability theory (discussed for instance in this previous blog post) is founded on the notion of a probability space {(\Omega, {\cal E}, {\bf P})} – a space {\Omega} (the sample space) equipped with a {\sigma}-algebra {{\cal E}} (the event space), together with a countably additive probability measure {{\bf P}: {\cal E} \rightarrow [0,1]} that assigns a real number in the interval {[0,1]} to each event.

    One can generalise the concept of a probability space to a finitely additive probability space, in which the event space {{\cal E}} is now only a Boolean algebra rather than a {\sigma}-algebra, and the measure {\mu} is now only finitely additive instead of countably additive, thus {{\bf P}( E \vee F ) = {\bf P}(E) + {\bf P}(F)} when {E,F} are disjoint events. By giving up countable additivity, one loses a fair amount of measure and integration theory, and in particular the notion of the expectation of a random variable becomes problematic (unless the random variable takes only finitely many values). Nevertheless, one can still perform a fair amount of probability theory in this weaker setting.

    In this post I would like to describe a further weakening of probability theory, which I will call qualitative probability theory, in which one does not assign a precise numerical probability value {{\bf P}(E)} to each event, but instead merely records whether this probability is zero, one, or something in between. Thus {{\bf P}} is now a function from {{\cal E}} to the set {\{0, I, 1\}}, where {I} is a new symbol that replaces all the elements of the open interval {(0,1)}. In this setting, one can no longer compute quantitative expressions, such as the mean or variance of a random variable; but one can still talk about whether an event holds almost surely, with positive probability, or with zero probability, and there are still usable notions of independence. (I will refer to classical probability theory as quantitative probability theory, to distinguish it from its qualitative counterpart.)

    The main reason I want to introduce this weak notion of probability theory is that it becomes suited to talk about random variables living inside algebraic varieties, even if these varieties are defined over fields other than {{\bf R}} or {{\bf C}}. In algebraic geometry one often talks about a “generic” element of a variety {V} defined over a field {k}, which does not lie in any specified variety of lower dimension defined over {k}. Once {V} has positive dimension, such generic elements do not exist as classical, deterministic {k}-points {x} in {V}, since of course any such point lies in the {0}-dimensional subvariety {\{x\}} of {V}. There are of course several established ways to deal with this problem. One way (which one might call the “Weil” approach to generic points) is to extend the field {k} to a sufficiently transcendental extension {\tilde k}, in order to locate a sufficient number of generic points in {V(\tilde k)}. Another approach (which one might dub the “Zariski” approach to generic points) is to work scheme-theoretically, and interpret a generic point in {V} as being associated to the zero ideal in the function ring of {V}. However I want to discuss a third perspective, in which one interprets a generic point not as a deterministic object, but rather as a random variable {{\bf x}} taking values in {V}, but which lies in any given lower-dimensional subvariety of {V} with probability zero. This interpretation is intuitive, but difficult to implement in classical probability theory (except perhaps when considering varieties over {{\bf R}} or {{\bf C}}) due to the lack of a natural probability measure to place on algebraic varieties; however it works just fine in qualitative probability theory. In particular, the algebraic geometry notion of being “generically true” can now be interpreted probabilistically as an assertion that something is “almost surely true”.

    It turns out that just as qualitative random variables may be used to interpret the concept of a generic point, they can also be used to interpret the concept of a type in model theory; the type of a random variable {x} is the set of all predicates {\phi(x)} that are almost surely obeyed by {x}. In contrast, model theorists often adopt a Weil-type approach to types, in which one works with deterministic representatives of a type, which often do not occur in the original structure of interest, but only in a sufficiently saturated extension of that structure (this is the analogue of working in a sufficiently transcendental extension of the base field). However, it seems that (in some cases at least) one can equivalently view types in terms of (qualitative) random variables on the original structure, avoiding the need to extend that structure. (Instead, one reserves the right to extend the sample space of one’s probability theory whenever necessary, as part of the “probabilistic way of thinking” discussed in this previous blog post.) We illustrate this below the fold with two related theorems that I will interpret through the probabilistic lens: the “group chunk theorem” of Weil (and later developed by Hrushovski), and the “group configuration theorem” of Zilber (and again later developed by Hrushovski). For sake of concreteness we will only consider these theorems in the theory of algebraically closed fields, although the results are quite general and can be applied to many other theories studied in model theory.

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    Let {F} be a field. A definable set over {F} is a set of the form

    \displaystyle  \{ x \in F^n | \phi(x) \hbox{ is true} \} \ \ \ \ \ (1)

    where {n} is a natural number, and {\phi(x)} is a predicate involving the ring operations {+,\times} of {F}, the equality symbol {=}, an arbitrary number of constants and free variables in {F}, the quantifiers {\forall, \exists}, boolean operators such as {\vee,\wedge,\neg}, and parentheses and colons, where the quantifiers are always understood to be over the field {F}. Thus, for instance, the set of quadratic residues

    \displaystyle  \{ x \in F | \exists y: x = y \times y \}

    is definable over {F}, and any algebraic variety over {F} is also a definable set over {F}. Henceforth we will abbreviate “definable over {F}” simply as “definable”.

    If {F} is a finite field, then every subset of {F^n} is definable, since finite sets are automatically definable. However, we can obtain a more interesting notion in this case by restricting the complexity of a definable set. We say that {E \subset F^n} is a definable set of complexity at most {M} if {n \leq M}, and {E} can be written in the form (1) for some predicate {\phi} of length at most {M} (where all operators, quantifiers, relations, variables, constants, and punctuation symbols are considered to have unit length). Thus, for instance, a hypersurface in {n} dimensions of degree {d} would be a definable set of complexity {O_{n,d}(1)}. We will then be interested in the regime where the complexity remains bounded, but the field size (or field characteristic) becomes large.

    In a recent paper, I established (in the large characteristic case) the following regularity lemma for dense definable graphs, which significantly strengthens the Szemerédi regularity lemma in this context, by eliminating “bad” pairs, giving a polynomially strong regularity, and also giving definability of the cells:

    Lemma 1 (Algebraic regularity lemma) Let {F} be a finite field, let {V,W} be definable non-empty sets of complexity at most {M}, and let {E \subset V \times W} also be definable with complexity at most {M}. Assume that the characteristic of {F} is sufficiently large depending on {M}. Then we may partition {V = V_1 \cup \ldots \cup V_m} and {W = W_1 \cup \ldots \cup W_n} with {m,n = O_M(1)}, with the following properties:

    • (Definability) Each of the {V_1,\ldots,V_m,W_1,\ldots,W_n} are definable of complexity {O_M(1)}.
    • (Size) We have {|V_i| \gg_M |V|} and {|W_j| \gg_M |W|} for all {i=1,\ldots,m} and {j=1,\ldots,n}.
    • (Regularity) We have

      \displaystyle  |E \cap (A \times B)| = d_{ij} |A| |B| + O_M( |F|^{-1/4} |V| |W| ) \ \ \ \ \ (2)

      for all {i=1,\ldots,m}, {j=1,\ldots,n}, {A \subset V_i}, and {B\subset W_j}, where {d_{ij}} is a rational number in {[0,1]} with numerator and denominator {O_M(1)}.

    My original proof of this lemma was quite complicated, based on an explicit calculation of the “square”

    \displaystyle  \mu(w,w') := \{ v \in V: (v,w), (v,w') \in E \}

    of {E} using the Lang-Weil bound and some facts about the étale fundamental group. It was the reliance on the latter which was the main reason why the result was restricted to the large characteristic setting. (I then applied this lemma to classify expanding polynomials over finite fields of large characteristic, but I will not discuss these applications here; see this previous blog post for more discussion.)

    Recently, Anand Pillay and Sergei Starchenko (and independently, Udi Hrushovski) have observed that the theory of the étale fundamental group is not necessary in the argument, and the lemma can in fact be deduced from quite general model theoretic techniques, in particular using (a local version of) the concept of stability. One of the consequences of this new proof of the lemma is that the hypothesis of large characteristic can be omitted; the lemma is now known to be valid for arbitrary finite fields {F} (although its content is trivial if the field is not sufficiently large depending on the complexity at most {M}).

    Inspired by this, I decided to see if I could find yet another proof of the algebraic regularity lemma, again avoiding the theory of the étale fundamental group. It turns out that the spectral proof of the Szemerédi regularity lemma (discussed in this previous blog post) adapts very nicely to this setting. The key fact needed about definable sets over finite fields is that their cardinality takes on an essentially discrete set of values. More precisely, we have the following fundamental result of Chatzidakis, van den Dries, and Macintyre:

    Proposition 2 Let {F} be a finite field, and let {M > 0}.

    • (Discretised cardinality) If {E} is a non-empty definable set of complexity at most {M}, then one has

      \displaystyle  |E| = c |F|^d + O_M( |F|^{d-1/2} ) \ \ \ \ \ (3)

      where {d = O_M(1)} is a natural number, and {c} is a positive rational number with numerator and denominator {O_M(1)}. In particular, we have {|F|^d \ll_M |E| \ll_M |F|^d}.

    • (Definable cardinality) Assume {|F|} is sufficiently large depending on {M}. If {V, W}, and {E \subset V \times W} are definable sets of complexity at most {M}, so that {E_w := \{ v \in V: (v,w) \in W \}} can be viewed as a definable subset of {V} that is definably parameterised by {w \in W}, then for each natural number {d = O_M(1)} and each positive rational {c} with numerator and denominator {O_M(1)}, the set

      \displaystyle  \{ w \in W: |E_w| = c |F|^d + O_M( |F|^{d-1/2} ) \} \ \ \ \ \ (4)

      is definable with complexity {O_M(1)}, where the implied constants in the asymptotic notation used to define (4) are the same as those that appearing in (3). (Informally: the “dimension” {d} and “measure” {c} of {E_w} depends definably on {w}.)

    We will take this proposition as a black box; a proof can be obtained by combining the description of definable sets over pseudofinite fields (discussed in this previous post) with the Lang-Weil bound (discussed in this previous post). (The former fact is phrased using nonstandard analysis, but one can use standard compactness-and-contradiction arguments to convert such statements to statements in standard analysis, as discussed in this post.)

    The above proposition places severe restrictions on the cardinality of definable sets; for instance, it shows that one cannot have a definable set of complexity at most {M} and cardinality {|F|^{1/2}}, if {|F|} is sufficiently large depending on {M}. If {E \subset V} are definable sets of complexity at most {M}, it shows that {|E| = (c+ O_M(|F|^{-1/2})) |V|} for some rational {0\leq c \leq 1} with numerator and denominator {O_M(1)}; furthermore, if {c=0}, we may improve this bound to {|E| = O_M( |F|^{-1} |V|)}. In particular, we obtain the following “self-improving” properties:

    • If {E \subset V} are definable of complexity at most {M} and {|E| \leq \epsilon |V|} for some {\epsilon>0}, then (if {\epsilon} is sufficiently small depending on {M} and {F} is sufficiently large depending on {M}) this forces {|E| = O_M( |F|^{-1} |V| )}.
    • If {E \subset V} are definable of complexity at most {M} and {||E| - c |V|| \leq \epsilon |V|} for some {\epsilon>0} and positive rational {c}, then (if {\epsilon} is sufficiently small depending on {M,c} and {F} is sufficiently large depending on {M,c}) this forces {|E| = c |V| + O_M( |F|^{-1/2} |V| )}.

    It turns out that these self-improving properties can be applied to the coefficients of various matrices (basically powers of the adjacency matrix associated to {E}) that arise in the spectral proof of the regularity lemma to significantly improve the bounds in that lemma; we describe how this is done below the fold. We also make some connections to the stability-based proofs of Pillay-Starchenko and Hrushovski.

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    The rectification principle in arithmetic combinatorics asserts, roughly speaking, that very small subsets (or, alternatively, small structured subsets) of an additive group or a field of large characteristic can be modeled (for the purposes of arithmetic combinatorics) by subsets of a group or field of zero characteristic, such as the integers {{\bf Z}} or the complex numbers {{\bf C}}. The additive form of this principle is known as the Freiman rectification principle; it has several formulations, going back of course to the original work of Freiman. Here is one formulation as given by Bilu, Lev, and Ruzsa:

    Proposition 1 (Additive rectification) Let {A} be a subset of the additive group {{\bf Z}/p{\bf Z}} for some prime {p}, and let {s \geq 1} be an integer. Suppose that {|A| \leq \log_{2s} p}. Then there exists a map {\phi: A \rightarrow A'} into a subset {A'} of the integers which is a Freiman isomorphism of order {s} in the sense that for any {x_1,\ldots,x_s,y_1,\ldots,y_s \in A}, one has

    \displaystyle  x_1+\ldots+x_s = y_1+\ldots+y_s

    if and only if

    \displaystyle  \phi(x_1)+\ldots+\phi(x_s) = \phi(y_1)+\ldots+\phi(y_s).

    Furthermore {\phi} is a right-inverse of the obvious projection homomorphism from {{\bf Z}} to {{\bf Z}/p{\bf Z}}.

    The original version of the rectification principle allowed the sets involved to be substantially larger in size (cardinality up to a small constant multiple of {p}), but with the additional hypothesis of bounded doubling involved; see the above-mentioned papers, as well as this later paper of Green and Ruzsa, for further discussion.

    The proof of Proposition 1 is quite short (see Theorem 3.1 of Bilu-Lev-Ruzsa); the main idea is to use Minkowski’s theorem to find a non-trivial dilate {aA} of {A} that is contained in a small neighbourhood of the origin in {{\bf Z}/p{\bf Z}}, at which point the rectification map {\phi} can be constructed by hand.

    Very recently, Codrut Grosu obtained an arithmetic analogue of the above theorem, in which the rectification map {\phi} preserves both additive and multiplicative structure:

    Theorem 2 (Arithmetic rectification) Let {A} be a subset of the finite field {{\bf F}_p} for some prime {p \geq 3}, and let {s \geq 1} be an integer. Suppose that {|A| < \log_2 \log_{2s} \log_{2s^2} p - 1}. Then there exists a map {\phi: A \rightarrow A'} into a subset {A'} of the complex numbers which is a Freiman field isomorphism of order {s} in the sense that for any {x_1,\ldots,x_n \in A} and any polynomial {P(x_1,\ldots,x_n)} of degree at most {s} and integer coefficients of magnitude summing to at most {s}, one has

    \displaystyle  P(x_1,\ldots,x_n)=0

    if and only if

    \displaystyle  P(\phi(x_1),\ldots,\phi(x_n))=0.

    Note that it is necessary to use an algebraically closed field such as {{\bf C}} for this theorem, in contrast to the integers used in Proposition 1, as {{\bf F}_p} can contain objects such as square roots of {-1} which can only map to {\pm i} in the complex numbers (once {s} is at least {2}).

    Using Theorem 2, one can transfer results in arithmetic combinatorics (e.g. sum-product or Szemerédi-Trotter type theorems) regarding finite subsets of {{\bf C}} to analogous results regarding sufficiently small subsets of {{\bf F}_p}; see the paper of Grosu for several examples of this. This should be compared with the paper of Vu, Wood, and Wood, which introduces a converse principle that embeds finite subsets of {{\bf C}} (or more generally, a characteristic zero integral domain) in a Freiman field-isomorphic fashion into finite subsets of {{\bf F}_p} for arbitrarily large primes {p}, allowing one to transfer arithmetic combinatorical facts from the latter setting to the former.

    Grosu’s argument uses some quantitative elimination theory, and in particular a quantitative variant of a lemma of Chang that was discussed previously on this blog. In that previous blog post, it was observed that (an ineffective version of) Chang’s theorem could be obtained using only qualitative algebraic geometry (as opposed to quantitative algebraic geometry tools such as elimination theory results with explicit bounds) by means of nonstandard analysis (or, in what amounts to essentially the same thing in this context, the use of ultraproducts). One can then ask whether one can similarly establish an ineffective version of Grosu’s result by nonstandard means. The purpose of this post is to record that this can indeed be done without much difficulty, though the result obtained, being ineffective, is somewhat weaker than that in Theorem 2. More precisely, we obtain

    Theorem 3 (Ineffective arithmetic rectification) Let {s, n \geq 1}. Then if {{\bf F}} is a field of characteristic at least {C_{s,n}} for some {C_{s,n}} depending on {s,n}, and {A} is a subset of {{\bf F}} of cardinality {n}, then there exists a map {\phi: A \rightarrow A'} into a subset {A'} of the complex numbers which is a Freiman field isomorphism of order {s}.

    Our arguments will not provide any effective bound on the quantity {C_{s,n}} (though one could in principle eventually extract such a bound by deconstructing the proof of Proposition 4 below), making this result weaker than Theorem 2 (save for the minor generalisation that it can handle fields of prime power order as well as fields of prime order as long as the characteristic remains large).

    Following the principle that ultraproducts can be used as a bridge to connect quantitative and qualitative results (as discussed in these previous blog posts), we will deduce Theorem 3 from the following (well-known) qualitative version:

    Proposition 4 (Baby Lefschetz principle) Let {k} be a field of characteristic zero that is finitely generated over the rationals. Then there is an isomorphism {\phi: k \rightarrow \phi(k)} from {k} to a subfield {\phi(k)} of {{\bf C}}.

    This principle (first laid out in an appendix of Lefschetz’s book), among other things, often allows one to use the methods of complex analysis (e.g. Riemann surface theory) to study many other fields of characteristic zero. There are many variants and extensions of this principle; see for instance this MathOverflow post for some discussion of these. I used this baby version of the Lefschetz principle recently in a paper on expanding polynomial maps.

    Proof: We give two proofs of this fact, one using transcendence bases and the other using Hilbert’s nullstellensatz.

    We begin with the former proof. As {k} is finitely generated over {{\bf Q}}, it has finite transcendence degree, thus one can find algebraically independent elements {x_1,\ldots,x_m} of {k} over {{\bf Q}} such that {k} is a finite extension of {{\bf Q}(x_1,\ldots,x_m)}, and in particular by the primitive element theorem {k} is generated by {{\bf Q}(x_1,\ldots,x_m)} and an element {\alpha} which is algebraic over {{\bf Q}(x_1,\ldots,x_m)}. (Here we use the fact that characteristic zero fields are separable.) If we then define {\phi} by first mapping {x_1,\ldots,x_m} to generic (and thus algebraically independent) complex numbers {z_1,\ldots,z_m}, and then setting {\phi(\alpha)} to be a complex root of of the minimal polynomial for {\alpha} over {{\bf Q}(x_1,\ldots,x_m)} after replacing each {x_i} with the complex number {z_i}, we obtain a field isomorphism {\phi: k \rightarrow \phi(k)} with the required properties.

    Now we give the latter proof. Let {x_1,\ldots,x_m} be elements of {k} that generate that field over {{\bf Q}}, but which are not necessarily algebraically independent. Our task is then equivalent to that of finding complex numbers {z_1,\ldots,z_m} with the property that, for any polynomial {P(x_1,\ldots,x_m)} with rational coefficients, one has

    \displaystyle  P(x_1,\ldots,x_m) = 0

    if and only if

    \displaystyle  P(z_1,\ldots,z_m) = 0.

    Let {{\mathcal P}} be the collection of all polynomials {P} with rational coefficients with {P(x_1,\ldots,x_m)=0}, and {{\mathcal Q}} be the collection of all polynomials {P} with rational coefficients with {P(x_1,\ldots,x_m) \neq 0}. The set

    \displaystyle  S := \{ (z_1,\ldots,z_m) \in {\bf C}^m: P(z_1,\ldots,z_m)=0 \hbox{ for all } P \in {\mathcal P} \}

    is the intersection of countably many algebraic sets and is thus also an algebraic set (by the Hilbert basis theorem or the Noetherian property of algebraic sets). If the desired claim failed, then {S} could be covered by the algebraic sets {\{ (z_1,\ldots,z_m) \in {\bf C}^m: Q(z_1,\ldots,z_m) = 0 \}} for {Q \in {\mathcal Q}}. By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over {{\bf C}} cannot be covered by countably many varieties of smaller dimension, we conclude that {S} must in fact be covered by a finite number of such sets, thus

    \displaystyle  S \subset \bigcup_{i=1}^n \{ (z_1,\ldots,z_m) \in {\bf C}^m: Q_i(z_1,\ldots,z_m) = 0 \}

    for some {Q_1,\ldots,Q_n \in {\bf C}^m}. By the nullstellensatz, we thus have an identity of the form

    \displaystyle  (Q_1 \ldots Q_n)^l = P_1 R_1 + \ldots + P_r R_r

    for some natural numbers {l,r \geq 1}, polynomials {P_1,\ldots,P_r \in {\mathcal P}}, and polynomials {R_1,\ldots,R_r} with coefficients in {\overline{{\bf Q}}}. In particular, this identity also holds in the algebraic closure {\overline{k}} of {k}. Evaluating this identity at {(x_1,\ldots,x_m)} we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows. \Box

    From Proposition 4 one can now deduce Theorem 3 by a routine ultraproduct argument (the same one used in these previous blog posts). Suppose for contradiction that Theorem 3 fails. Then there exists natural numbers {s,n \geq 1}, a sequence of finite fields {{\bf F}_i} of characteristic at least {i}, and subsets {A_i=\{a_{i,1},\ldots,a_{i,n}\}} of {{\bf F}_i} of cardinality {n} such that for each {i}, there does not exist a Freiman field isomorphism of order {s} from {A_i} to the complex numbers. Now we select a non-principal ultrafilter {\alpha \in \beta {\bf N} \backslash {\bf N}}, and construct the ultraproduct {{\bf F} := \prod_{i \rightarrow \alpha} {\bf F}_i} of the finite fields {{\bf F}_i}. This is again a field (and is a basic example of what is known as a pseudo-finite field); because the characteristic of {{\bf F}_i} goes to infinity as {i \rightarrow \infty}, it is easy to see (using Los’s theorem) that {{\bf F}} has characteristic zero and can thus be viewed as an extension of the rationals {{\bf Q}}.

    Now let {a_j := \lim_{i \rightarrow \alpha} a_{i,j}} be the ultralimit of the {a_{i,j}}, so that {A := \{a_1,\ldots,a_n\}} is the ultraproduct of the {A_i}, then {A} is a subset of {{\bf F}} of cardinality {n}. In particular, if {k} is the field generated by {{\bf Q}} and {A}, then {k} is a finitely generated extension of the rationals and thus, by Proposition 4 there is an isomorphism {\phi: k \rightarrow \phi(k)} from {k} to a subfield {\phi(k)} of the complex numbers. In particular, {\phi(a_1),\ldots,\phi(a_n)} are complex numbers, and for any polynomial {P(x_1,\ldots,x_n)} with integer coefficients, one has

    \displaystyle  P(a_1,\ldots,a_n) = 0

    if and only if

    \displaystyle  P(\phi(a_1),\ldots,\phi(a_n)) = 0.

    By Los’s theorem, we then conclude that for all {i} sufficiently close to {\alpha}, one has for all polynomials {P(x_1,\ldots,x_n)} of degree at most {s} and whose coefficients are integers whose magnitude sums up to {s}, one has

    \displaystyle  P(a_{i,1},\ldots,a_{i,n}) = 0

    if and only if

    \displaystyle  P(\phi(a_1),\ldots,\phi(a_n)) = 0.

    But this gives a Freiman field isomorphism of order {s} between {A_i} and {\phi(A)}, contradicting the construction of {A_i}, and Theorem 3 follows.

    Two weeks ago I was at Oberwolfach, for the Arbeitsgemeinschaft in Ergodic Theory and Combinatorial Number Theory that I was one of the organisers for. At this workshop, I learned the details of a very nice recent convergence result of Miguel Walsh (who, incidentally, is an informal grandstudent of mine, as his advisor, Roman Sasyk, was my informal student), which considerably strengthens and generalises a number of previous convergence results in ergodic theory (including one of my own), with a remarkably simple proof. Walsh’s argument is phrased in a finitary language (somewhat similar, in fact, to the approach used in my paper mentioned previously), and (among other things) relies on the concept of metastability of sequences, a variant of the notion of convergence which is useful in situations in which one does not expect a uniform convergence rate; see this previous blog post for some discussion of metastability. When interpreted in a finitary setting, this concept requires a fair amount of “epsilon management” to manipulate; also, Walsh’s argument uses some other epsilon-intensive finitary arguments, such as a decomposition lemma of Gowers based on the Hahn-Banach theorem. As such, I was tempted to try to rewrite Walsh’s argument in the language of nonstandard analysis to see the extent to which these sorts of issues could be managed. As it turns out, the argument gets cleaned up rather nicely, with the notion of metastability being replaced with the simpler notion of external Cauchy convergence (which we will define below the fold).

    Let’s first state Walsh’s theorem. This theorem is a norm convergence theorem in ergodic theory, and can be viewed as a substantial generalisation of one of the most fundamental theorems of this type, namely the mean ergodic theorem:

    Theorem 1 (Mean ergodic theorem) Let {(X,\mu,T)} be a measure-preserving system (a probability space {(X,\mu)} with an invertible measure-preserving transformation {T}). Then for any {f \in L^2(X,\mu)}, the averages {\frac{1}{N} \sum_{n=1}^N T^n f} converge in {L^2(X,\mu)} norm as {N \rightarrow \infty}, where {T^n f(x) := f(T^{-n} x)}.

    In this post, all functions in {L^2(X,\mu)} and similar spaces will be taken to be real instead of complex-valued for simplicity, though the extension to the complex setting is routine.

    Actually, we have a precise description of the limit of these averages, namely the orthogonal projection of {f} to the {T}-invariant factors. (See for instance my lecture notes on this theorem.) While this theorem ostensibly involves measure theory, it can be abstracted to the more general setting of unitary operators on a Hilbert space:

    Theorem 2 (von Neumann mean ergodic theorem) Let {H} be a Hilbert space, and let {U: H \rightarrow H} be a unitary operator on {H}. Then for any {f \in H}, the averages {\frac{1}{N} \sum_{n=1}^N U^n f} converge strongly in {H} as {N \rightarrow \infty}.

    Again, see my lecture notes (or just about any text in ergodic theory) for a proof.

    Now we turn to Walsh’s theorem.

    Theorem 3 (Walsh’s convergence theorem) Let {(X,\mu)} be a measure space with a measure-preserving action of a nilpotent group {G}. Let {g_1,\ldots,g_k: {\bf Z} \rightarrow G} be polynomial sequences in {G} (i.e. each {g_i} takes the form {g_i(n) = a_{i,1}^{p_{i,1}(n)} \ldots a_{i,j}^{p_{i,j}(n)}} for some {a_{i,1},\ldots,a_{i,j} \in G} and polynomials {p_{i,1},\ldots,p_{i,j}: {\bf Z} \rightarrow {\bf Z}}). Then for any {f_1,\ldots,f_k \in L^\infty(X,\mu)}, the averages {\frac{1}{N} \sum_{n=1}^N (g_1(n) f_1) \ldots (g_k(n) f_k)} converge in {L^2(X,\mu)} norm as {N \rightarrow \infty}, where {g(n) f(x) := f(g(n)^{-1} x)}.

    It turns out that this theorem can also be abstracted to some extent, although due to the multiplication in the summand {(g_1(n) f_1) \ldots (g_k(n) f_k)}, one cannot work purely with Hilbert spaces as in the von Neumann mean ergodic theorem, but must also work with something like the Banach algebra {L^\infty(X,\mu)}. There are a number of ways to formulate this abstraction (which will be of some minor convenience to us, as it will allow us to reduce the need to invoke the nonstandard measure theory of Loeb, discussed for instance in this blog post); we will use the notion of a (real) commutative probability space {({\mathcal A},\tau)}, which for us will be a commutative unital algebra {{\mathcal A}} over the reals together with a linear functional {\tau: {\mathcal A} \rightarrow {\bf R}} which maps {1} to {1} and obeys the non-negativity axiom {\tau(f^2) \ge 0} for all {f}. The key example to keep in mind here is {{\mathcal A} = L^\infty(X,\mu)} of essentially bounded real-valued measurable functions with the supremum norm, and with the trace {\tau(f) := \int_X f\ d\mu}. We will also assume in our definition of commutative probability spaces that all elements {f} of {{\mathcal A}} are bounded in the sense that the spectral radius {\rho(f) := \lim_{k \rightarrow \infty} \tau(f^{2k})^{1/2k}} is finite. (In the concrete case of {L^\infty(X,\mu)}, the spectral radius is just the {L^\infty} norm.)

    Given a commutative probability space, we can form an inner product {\langle, \rangle_{L^2(\tau)}} on it by the formula

    \displaystyle  \langle f, g \rangle_{L^2(\tau)} := \tau(fg).

    This is a positive semi-definite form, and gives a (possibly degenerate) inner product structure on {{\mathcal A}}. We could complete this structure into a Hilbert space {L^2(\tau)} (after quotienting out the elements of zero norm), but we will not do so here, instead just viewing {L^2(\tau)} as providing a semi-metric on {{\mathcal A}}. For future reference we record the inequalities

    \displaystyle  \rho(fg) \leq \rho(f) \rho(g)

    \displaystyle  \rho(f+g) \leq \rho(f) + \rho(g)

    \displaystyle  \| fg\|_{L^2(\tau)} \leq \|f\|_{L^2(\tau)} \rho(g)

    for any {f,g}, which we will use in the sequel without further comment; see e.g. these previous blog notes for proofs. (Actually, for the purposes of proving Theorem 3, one can specialise to the {L^\infty(X,\mu)} case (and ultraproducts thereof), in which case these inequalities are just the triangle and Hölder inequalities.)

    The abstract version of Theorem 3 is then

    Theorem 4 (Walsh’s theorem, abstract version) Let {({\mathcal A},\tau)} be a commutative probability space, and let {G} be a nilpotent group acting on {{\mathcal A}} by isomorphisms (preserving the algebra, conjugation, and trace structure, and thus also preserving the spectral radius and {L^2(\tau)} norm). Let {g_1,\ldots,g_k: {\bf Z} \rightarrow G} be polynomial sequences. Then for any {f_1,\ldots,f_k \in {\mathcal A}}, the averages {\frac{1}{N} \sum_{n=1}^N (g_1(n) f_1) \ldots (g_k(n) f_k)} form a Cauchy sequence in {L^2(\tau)} (semi-)norm as {N \rightarrow \infty}.

    It is easy to see that this theorem generalises Theorem 3. Conversely, one can use the commutative Gelfand-Naimark theorem to deduce Theorem 4 from Theorem 3, although we will not need this implication. Note how we are abandoning all attempts to discern what the limit of the sequence actually is, instead contenting ourselves with demonstrating that it is merely a Cauchy sequence. With this phrasing, it is tempting to ask whether there is any analogue of Walsh’s theorem for noncommutative probability spaces, but unfortunately the answer to that question is negative for all but the simplest of averages, as was worked out in this paper of Austin, Eisner, and myself.

    Our proof of Theorem 4 will proceed as follows. Firstly, in order to avoid the epsilon management alluded to earlier, we will take an ultraproduct to rephrase the theorem in the language of nonstandard analysis; for reasons that will be clearer later, we will also convert the convergence problem to a problem of obtaining metastability (external Cauchy convergence). Then, we observe that (the nonstandard counterpart of) the expression {\|\frac{1}{N} \sum_{n=1}^N (g_1(n) f_1) \ldots (g_k(n) f_k)\|_{L^2(\tau)}^2} can be viewed as the inner product of (say) {f_k} with a certain type of expression, which we call a dual function. By performing an orthogonal projection to the span of the dual functions, we can split {f_k} into the sum of an expression orthogonal to all dual functions (the “pseudorandom” component), and a function that can be well approximated by finite linear combinations of dual functions (the “structured” component). The contribution of the pseudorandom component is asymptotically negligible, so we can reduce to consideration of the structured component. But by a little bit of rearrangement, this can be viewed as an average of expressions similar to the initial average {\frac{1}{N} \sum_{n=1}^N (g_1(n) f_1) \ldots (g_k(n) f_k)}, except with the polynomials {g_1,\ldots,g_k} replaced by a “lower complexity” set of such polynomials, which can be greater in number, but which have slightly lower degrees in some sense. One can iterate this (using “PET induction”) until all the polynomials become trivial, at which point the claim follows.

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