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I’ve just uploaded to the arXiv my paper “Inverse theorems for sets and measures of polynomial growth“. This paper was motivated by two related questions. The first question was to obtain a qualitatively precise description of the sets of polynomial growth that arise in Gromov’s theorem, in much the same way that Freiman’s theorem (and its generalisations) provide a qualitatively precise description of sets of small doubling. The other question was to obtain a non-abelian analogue of inverse Littlewood-Offord theory.

Let me discuss the former question first. Gromov’s theorem tells us that if a finite subset ${A}$ of a group ${G}$ exhibits polynomial growth in the sense that ${|A^n|}$ grows polynomially in ${n}$, then the group generated by ${A}$ is virtually nilpotent (the converse direction also true, and is relatively easy to establish). This theorem has been strengthened a number of times over the years. For instance, a few years ago, I proved with Shalom that the condition that ${|A^n|}$ grew polynomially in ${n}$ could be replaced by ${|A^n| \leq C n^d}$ for a single ${n}$, as long as ${n}$ was sufficiently large depending on ${C,d}$ (in fact we gave a fairly explicit quantitative bound on how large ${n}$ needed to be). A little more recently, with Breuillard and Green, the condition ${|A^n| \leq C n^d}$ was weakened to ${|A^n| \leq n^d |A|}$, that is to say it sufficed to have polynomial relative growth at a finite scale. In fact, the latter paper gave more information on ${A}$ in this case, roughly speaking it showed (at least in the case when ${A}$ was a symmetric neighbourhood of the identity) that ${A^n}$ was “commensurate” with a very structured object known as a coset nilprogression. This can then be used to establish further control on ${A}$. For instance, it was recently shown by Breuillard and Tointon (again in the symmetric case) that if ${|A^n| \leq n^d |A|}$ for a single ${n}$ that was sufficiently large depending on ${d}$, then all the ${A^{n'}}$ for ${n' \geq n}$ have a doubling constant bounded by a bound ${C_d}$ depending only on ${d}$, thus ${|A^{2n'}| \leq C_d |A^{n'}|}$ for all ${n' \geq n}$.

In this paper we are able to refine this analysis a bit further; under the same hypotheses, we can show an estimate of the form

$\displaystyle \log |A^{n'}| = \log |A^n| + f( \log n' - \log n ) + O_d(1)$

for all ${n' \geq n}$ and some piecewise linear, continuous, non-decreasing function ${f: [0,+\infty) \rightarrow [0,+\infty)}$ with ${f(0)=0}$, where the error ${O_d(1)}$ is bounded by a constant depending only on ${d}$, and where ${f}$ has at most ${O_d(1)}$ pieces, each of which has a slope that is a natural number of size ${O_d(1)}$. To put it another way, the function ${n' \mapsto |A^{n'}|}$ for ${n' \geq n}$ behaves (up to multiplicative constants) like a piecewise polynomial function, where the degree of the function and number of pieces is bounded by a constant depending on ${d}$.

One could ask whether the function ${f}$ has any convexity or concavity properties. It turns out that it can exhibit either convex or concave behaviour (or a combination of both). For instance, if ${A}$ is contained in a large finite group, then ${n \mapsto |A^n|}$ will eventually plateau to a constant, exhibiting concave behaviour. On the other hand, in nilpotent groups one can see convex behaviour; for instance, in the Heisenberg group ${\begin{pmatrix}{} {1} {\mathbf Z} {\mathbf Z} \\ {0} {1} {\mathbf Z} \\ {0} {1} \end{pmatrix}}$, if one sets ${A}$ to be a set of matrices of the form ${\begin{pmatrix} 1 & O(N) & O(N^3) \\ 0 & 1 & O(N) \\ 0 & 0 & 1 \end{pmatrix}}$ for some large ${N}$ (abusing the ${O()}$ notation somewhat), then ${n \mapsto A^n}$ grows cubically for ${n \leq N}$ but then grows quartically for ${n > N}$.

To prove this proposition, it turns out (after using a somewhat difficult inverse theorem proven previously by Breuillard, Green, and myself) that one has to analyse the volume growth ${n \mapsto |P^n|}$ of nilprogressions ${P}$. In the “infinitely proper” case where there are no unexpected relations between the generators of the nilprogression, one can lift everything to a simply connected Lie group (where one can take logarithms and exploit the Baker-Campbell-Hausdorff formula heavily), eventually describing ${P^n}$ with fair accuracy by a certain convex polytope with vertices depending polynomially on ${n}$, which implies that ${|P^n|}$ depends polynomially on ${n}$ up to constants. If one is not in the “infinitely proper” case, then at some point ${n_0}$ the nilprogression ${P^{n_0}}$ develops a “collision”, but then one can use this collision to show (after some work) that the dimension of the “Lie model” of ${P^{n_0}}$ has dropped by at least one from the dimension of ${P}$ (the notion of a Lie model being developed in the previously mentioned paper of Breuillard, Greenm, and myself), so that this sort of collision can only occur a bounded number of times, with essentially polynomial volume growth behaviour between these collisions.

The arguments also give a precise description of the location of a set ${A}$ for which ${A^n}$ grows polynomially in ${n}$. In the symmetric case, what ends up happening is that ${A^n}$ becomes commensurate to a “coset nilprogression” ${HP}$ of bounded rank and nilpotency class, whilst ${A}$ is “virtually” contained in a scaled down version ${HP^{1/n}}$ of that nilprogression. What “virtually” means is a little complicated; roughly speaking, it means that there is a set ${X}$ of bounded cardinality such that ${aXHP^{1/n} \approx XHP^{1/n}}$ for all ${a \in A}$. Conversely, if ${A}$ is virtually contained in ${HP^{1/n}}$, then ${A^n}$ is commensurate to ${HP}$ (and more generally, ${A^{mn}}$ is commensurate to ${HP^m}$ for any natural number ${m}$), giving quite a (qualitatively) precise description of ${A}$ in terms of coset nilprogressions.

The main tool used to prove these results is the structure theorem for approximate groups established by Breuillard, Green, and myself, which roughly speaking asserts that approximate groups are always commensurate with coset nilprogressions. A key additional trick is a pigeonholing argument of Sanders, which in this context is the assertion that if ${A^n}$ is comparable to ${A^{2n}}$, then there is an ${n'}$ between ${n}$ and ${2n}$ such that ${A \cdot A^{n'}}$ is very close in size to ${A^{n'}}$ (up to a relative error of ${1/n}$). It is this fact, together with the comparability of ${A^{n'}}$ to a coset nilprogression ${HP}$, that allows us (after some combinatorial argument) to virtually place ${A}$ inside ${HP^{1/n}}$.

Similar arguments apply when discussing iterated convolutions ${\mu^{*n}}$ of (symmetric) probability measures on a (discrete) group ${G}$, rather than combinatorial powers ${A^n}$ of a finite set. Here, the analogue of volume ${A^n}$ is given by the negative power ${\| \mu^{*n} \|_{\ell^2}^{-2}}$ of the ${\ell^2}$ norm of ${\mu^{*n}}$ (thought of as a non-negative function on ${G}$ of total mass 1). One can also work with other norms here than ${\ell^2}$, but this norm has some minor technical conveniences (and other measures of the “spread” of ${\mu^{*n}}$ end up being more or less equivalent for our purposes). There is an analogous structure theorem that asserts that if ${\mu^{*n}}$ spreads at most polynomially in ${n}$, then ${\mu^{*n}}$ is “commensurate” with the uniform probability distribution on a coset progression ${HP}$, and ${\mu}$ itself is largely concentrated near ${HP^{1/\sqrt{n}}}$. The factor of ${\sqrt{n}}$ here is the familiar scaling factor in random walks that arises for instance in the central limit theorem. The proof of (the precise version of) this statement proceeds similarly to the combinatorial case, using pigeonholing to locate a scale ${n'}$ where ${\mu *\mu^{n'}}$ has almost the same ${\ell^2}$ norm as ${\mu^{n'}}$.

A special case of this theory occurs when ${\mu}$ is the uniform probability measure on ${n}$ elements ${v_1,\dots,v_n}$ of ${G}$ and their inverses. The probability measure ${\mu^{*n}}$ is then the distribution of a random product ${w_1 \dots w_n}$, where each ${w_i}$ is equal to one of ${v_{j_i}}$ or its inverse ${v_{j_i}^{-1}}$, selected at random with ${j_i}$ drawn uniformly from ${\{1,\dots,n\}}$ with replacement. This is very close to the Littlewood-Offord situation of random products ${u_1 \dots u_n}$ where each ${u_i}$ is equal to ${v_i}$ or ${v_i^{-1}}$ selected independently at random (thus ${j_i}$ is now fixed to equal ${i}$ rather than being randomly drawn from ${\{1,\dots,n\}}$. In the case when ${G}$ is abelian, it turns out that a little bit of Fourier analysis shows that these two random walks have “comparable” distributions in a certain ${\ell^2}$ sense. As a consequence, the results in this paper can be used to recover an essentially optimal abelian inverse Littlewood-Offord theorem of Nguyen and Vu. In the nonabelian case, the only Littlewood-Offord theorem I am aware of is a recent result of Tiep and Vu for matrix groups, but in this case I do not know how to relate the above two random walks to each other, and so we can only obtain an analogue of the Tiep-Vu results for the symmetrised random walk ${w_1 \dots w_n}$ instead of the ordered random walk ${u_1 \dots u_n}$.

Suppose that ${A \subset B}$ are two subgroups of some ambient group ${G}$, with the index ${K := [B:A]}$ of ${A}$ in ${B}$ being finite. Then ${B}$ is the union of ${K}$ left cosets of ${A}$, thus ${B = SA}$ for some set ${S \subset B}$ of cardinality ${K}$. The elements ${s}$ of ${S}$ are not entirely arbitrary with regards to ${A}$. For instance, if ${A}$ is a normal subgroup of ${B}$, then for each ${s \in S}$, the conjugation map ${g \mapsto s^{-1} g s}$ preserves ${A}$. In particular, if we write ${A^s := s^{-1} A s}$ for the conjugate of ${A}$ by ${s}$, then

$\displaystyle A = A^s.$

Even if ${A}$ is not normal in ${B}$, it turns out that the conjugation map ${g \mapsto s^{-1} g s}$ approximately preserves ${A}$, if ${K}$ is bounded. To quantify this, let us call two subgroups ${A,B}$ ${K}$-commensurate for some ${K \geq 1}$ if one has

$\displaystyle [A : A \cap B], [B : A \cap B] \leq K.$

Proposition 1 Let ${A \subset B}$ be groups, with finite index ${K = [B:A]}$. Then for every ${s \in B}$, the groups ${A}$ and ${A^s}$ are ${K}$-commensurate, in fact

$\displaystyle [A : A \cap A^s ] = [A^s : A \cap A^s ] \leq K.$

Proof: One can partition ${B}$ into ${K}$ left translates ${xA}$ of ${A}$, as well as ${K}$ left translates ${yA^s}$ of ${A^s}$. Combining the partitions, we see that ${B}$ can be partitioned into at most ${K^2}$ non-empty sets of the form ${xA \cap yA^s}$. Each of these sets is easily seen to be a left translate of the subgroup ${A \cap A^s}$, thus ${[B: A \cap A^s] \leq K^2}$. Since

$\displaystyle [B: A \cap A^s] = [B:A] [A: A \cap A^s] = [B:A^s] [A^s: A \cap A^s]$

and ${[B:A] = [B:A^s]=K}$, we obtain the claim. $\Box$

One can replace the inclusion ${A \subset B}$ by commensurability, at the cost of some worsening of the constants:

Corollary 2 Let ${A, B}$ be ${K}$-commensurate subgroups of ${G}$. Then for every ${s \in B}$, the groups ${A}$ and ${A^s}$ are ${K^2}$-commensurate.

Proof: Applying the previous proposition with ${A}$ replaced by ${A \cap B}$, we see that for every ${s \in B}$, ${A \cap B}$ and ${(A \cap B)^s}$ are ${K}$-commensurate. Since ${A \cap B}$ and ${(A \cap B)^s}$ have index at most ${K}$ in ${A}$ and ${A^s}$ respectively, the claim follows. $\Box$

It turns out that a similar phenomenon holds for the more general concept of an approximate group, and gives a “classification” of all the approximate groups ${B}$ containing a given approximate group ${A}$ as a “bounded index approximate subgroup”. Recall that a ${K}$-approximate group ${A}$ in a group ${G}$ for some ${K \geq 1}$ is a symmetric subset of ${G}$ containing the identity, such that the product set ${A^2 := \{ a_1 a_2: a_1,a_2 \in A\}}$ can be covered by at most ${K}$ left translates of ${A}$ (and thus also ${K}$ right translates, by the symmetry of ${A}$). For simplicity we will restrict attention to finite approximate groups ${A}$ so that we can use their cardinality ${A}$ as a measure of size. We call finite two approximate groups ${A,B}$ ${K}$-commensurate if one has

$\displaystyle |A^2 \cap B^2| \geq \frac{1}{K} |A|, \frac{1}{K} |B|;$

note that this is consistent with the previous notion of commensurability for genuine groups.

Theorem 3 Let ${G}$ be a group, and let ${K_1,K_2,K_3 \geq 1}$ be real numbers. Let ${A}$ be a finite ${K_1}$-approximate group, and let ${B}$ be a symmetric subset of ${G}$ that contains ${A}$.

• (i) If ${B}$ is a ${K_2}$-approximate group with ${|B| \leq K_3 |A|}$, then one has ${B \subset SA}$ for some set ${S}$ of cardinality at most ${K_1 K_2 K_3}$. Furthermore, for each ${s \in S}$, the approximate groups ${A}$ and ${A^s}$ are ${K_1 K_2^5 K_3}$-commensurate.
• (ii) Conversely, if ${B \subset SA}$ for some set ${S}$ of cardinality at most ${K_3}$, and ${A}$ and ${A^s}$ are ${K_2}$-commensurate for all ${s \in S}$, then ${|B| \leq K_3 |A|}$, and ${B}$ is a ${K_1^6 K_2 K_3^2}$-approximate group.

Informally, the assertion that ${B}$ is an approximate group containing ${A}$ as a “bounded index approximate subgroup” is equivalent to ${B}$ being covered by a bounded number of shifts ${sA}$ of ${A}$, where ${s}$ approximately normalises ${A^2}$ in the sense that ${A^2}$ and ${(A^2)^s}$ have large intersection. Thus, to classify all such ${B}$, the problem essentially reduces to that of classifying those ${s}$ that approximately normalise ${A^2}$.

To prove the theorem, we recall some standard lemmas from arithmetic combinatorics, which are the foundation stones of the “Ruzsa calculus” that we will use to establish our results:

Lemma 4 (Ruzsa covering lemma) If ${A}$ and ${B}$ are finite non-empty subsets of ${G}$, then one has ${B \subset SAA^{-1}}$ for some set ${S \subset B}$ with cardinality ${|S| \leq \frac{|BA|}{|A|}}$.

Proof: We take ${S}$ to be a subset of ${B}$ with the property that the translates ${sA, s \in S}$ are disjoint in ${BA}$, and such that ${S}$ is maximal with respect to set inclusion. The required properties of ${S}$ are then easily verified. $\Box$

Lemma 5 (Ruzsa triangle inequality) If ${A,B,C}$ are finite non-empty subsets of ${G}$, then

$\displaystyle |A C^{-1}| \leq |A B^{-1}| |B C^{-1}| / |B|.$

Proof: If ${ac^{-1}}$ is an element of ${AC^{-1}}$ with ${a \in A}$ and ${c \in C}$, then from the identity ${ac^{-1} = (ab^{-1}) (bc^{-1})}$ we see that ${ac^{-1}}$ can be written as the product of an element of ${AB^{-1}}$ and an element of ${BC^{-1}}$ in at least ${|B|}$ distinct ways. The claim follows. $\Box$

Now we can prove (i). By the Ruzsa covering lemma, ${B}$ can be covered by at most

$\displaystyle \frac{|BA|}{|A|} \leq \frac{|B^2|}{|A|} \leq \frac{K_2 |B|}{|A|} \leq K_2 K_3$

left-translates of ${A^2}$, and hence by at most ${K_1 K_2 K_3}$ left-translates of ${A}$, thus ${B \subset SA}$ for some ${|S| \leq K_1 K_2 K_3}$. Since ${sA}$ only intersects ${B}$ if ${s \in BA}$, we may assume that

$\displaystyle S \subset BA \subset B^2$

and hence for any ${s \in S}$

$\displaystyle |A^s A| \leq |B^2 A B^2 A| \leq |B^6|$

$\displaystyle \leq K_2^5 |B| \leq K_2^5 K_3 |A|.$

By the Ruzsa covering lemma again, this implies that ${A^s}$ can be covered by at most ${K_2^5 K_3}$ left-translates of ${A^2}$, and hence by at most ${K_1 K_2^5 K_3}$ left-translates of ${A}$. By the pigeonhole principle, there thus exists a group element ${g}$ with

$\displaystyle |A^s \cap gA| \geq \frac{1}{K_1 K_2^5 K_3} |A|.$

Since

$\displaystyle |A^s \cap gA| \leq | (A^s \cap gA)^{-1} (A^s \cap gA)|$

and

$\displaystyle (A^s \cap gA)^{-1} (A^s \cap gA) \subset A^2 \cap (A^s)^2$

the claim follows.

Now we prove (ii). Clearly

$\displaystyle |B| \leq |S| |A| \leq K_3 |A|.$

Now we control the size of ${B^2 A}$. We have

$\displaystyle |B^2 A| \leq |SA SA^2| \leq K_3^2 \sup_{s \in S} |A s A^2| = K_3^2 \sup_{s \in S} |A^s A^2|.$

From the Ruzsa triangle inequality and symmetry we have

$\displaystyle |A^s A^2| \leq \frac{ |A^s (A^2 \cap (A^2)^s)| |(A^2 \cap (A^2)^s) A^2|}{|A^2 \cap (A^2)^s|}$

$\displaystyle \leq \frac{ |(A^3)^s| |A^4| }{|A|/K_2}$

$\displaystyle \leq K_2 \frac{ |A^3| |A^4| }{|A|}$

$\displaystyle \leq K_1^5 K_2 |A|$

and thus

$\displaystyle |B^2 A| \leq K_1^5 K_2 K_3^2 |A|.$

By the Ruzsa covering lemma, this implies that ${B^2}$ is covered by at most ${K_1^5 K_2 K_3^2}$ left-translates of ${A^2}$, hence by at most ${K_1^6 K_2 K_3^2}$ left-translates of ${A}$. Since ${A \subset B}$, the claim follows.

We now establish some auxiliary propositions about commensurability of approximate groups. The first claim is that commensurability is approximately transitive:

Proposition 6 Let ${A}$ be a ${K_1}$-approximate group, ${B}$ be a ${K_2}$-approximate group, and ${C}$ be a ${K_3}$-approximate group. If ${A}$ and ${B}$ are ${K_4}$-commensurate, and ${B}$ and ${C}$ are ${K_5}$-commensurate, then ${A}$ and ${C}$ are ${K_1^2 K_2^3 K_2^3 K_4 K_5 \max(K_1,K_3)}$-commensurate.

Proof: From two applications of the Ruzsa triangle inequality we have

$\displaystyle |AC| \leq \frac{|A (A^2 \cap B^2)| |(A^2 \cap B^2) (B^2 \cap C^2)| |(B^2 \cap C^2) C|}{|A^2 \cap B^2| |B^2 \cap C^2|}$

$\displaystyle \leq \frac{|A^3| |B^4| |C^3|}{ (|A|/K_4) (|B|/K_5)}$

$\displaystyle \leq K_4 K_5 \frac{K_1^2 |A| K_2^3 |B| K_3^2 |C|}{ |A| |B| }$

$\displaystyle = K_1^2 K_2^3 K_3^2 K_4 K_5 |C|.$

By the Ruzsa covering lemma, we may thus cover ${A}$ by at most ${K_1^2 K_2^3 K_3^2 K_4 K_5}$ left-translates of ${C^2}$, and hence by ${K_1^2 K_2^3 K_3^3 K_4 K_5}$ left-translates of ${C}$. By the pigeonhole principle, there thus exists a group element ${g}$ such that

$\displaystyle |A \cap gC| \geq \frac{1}{K_1^2 K_2^3 K_3^3 K_4 K_5} |A|,$

and so by arguing as in the proof of part (i) of the theorem we have

$\displaystyle |A^2 \cap C^2| \geq \frac{1}{K_1^2 K_2^3 K_3^3 K_4 K_5} |A|$

and similarly

$\displaystyle |A^2 \cap C^2| \geq \frac{1}{K_1^3 K_2^3 K_3^2 K_4 K_5} |C|$

and the claim follows. $\Box$

The next proposition asserts that the union and (modified) intersection of two commensurate approximate groups is again an approximate group:

Proposition 7 Let ${A}$ be a ${K_1}$-approximate group, ${B}$ be a ${K_2}$-approximate group, and suppose that ${A}$ and ${B}$ are ${K_3}$-commensurate. Then ${A \cup B}$ is a ${K_1 + K_2 + K_1^2 K_2^4 K_3 + K_1^4 K_2^2 K_3}$-approximate subgroup, and ${A^2 \cap B^2}$ is a ${K_1^6 K_2^3 K_3}$-approximate subgroup.

Using this proposition, one may obtain a variant of the previous theorem where the containment ${A \subset B}$ is replaced by commensurability; we leave the details to the interested reader.

Proof: We begin with ${A \cup B}$. Clearly ${A \cup B}$ is symmetric and contains the identity. We have ${(A \cup B)^2 = A^2 \cup AB \cup BA \cup B^2}$. The set ${A^2}$ is already covered by ${K_1}$ left translates of ${A}$, and hence of ${A \cup B}$; similarly ${B^2}$ is covered by ${K_2}$ left translates of ${A \cup B}$. As for ${AB}$, we observe from the Ruzsa triangle inequality that

$\displaystyle |AB^2| \leq \frac{|A (A^2 \cap B^2)| |(A^2 \cap B^2) B^2|}{|A^2 \cap B^2|}$

$\displaystyle \leq \frac{|A^3| |B^4|}{|A|/K_3}$

$\displaystyle \leq K_1^2 K_2^3 K_3 |B|$

and hence by the Ruzsa covering lemma, ${AB}$ is covered by at most ${K_1^2 K_2^3 K_3}$ left translates of ${B^2}$, and hence by ${K_1^2 K_2^4 K_3}$ left translates of ${B}$, and hence of ${A \cup B}$. Similarly ${BA}$ is covered by at most ${K_1^4 K_2^2 K_3}$ left translates of ${B}$. The claim follows.

Now we consider ${A^2 \cap B^2}$. Again, this is clearly symmetric and contains the identity. Repeating the previous arguments, we see that ${A}$ is covered by at most ${K_1^2 K_2^3 K_3}$ left-translates of ${B}$, and hence there exists a group element ${g}$ with

$\displaystyle |A \cap gB| \geq \frac{1}{K_1^2 K_2^3 K_3} |A|.$

Now observe that

$\displaystyle |(A^2 \cap B^2)^2 (A \cap gB)| \leq |A^5| \leq K_1^4 |A|$

and so by the Ruzsa covering lemma, ${(A^2 \cap B^2)^2}$ can be covered by at most ${K_1^6 K_2^3 K_3}$ left-translates of ${(A \cap gB) (A \cap gB)^{-1}}$. But this latter set is (as observed previously) contained in ${A^2 \cap B^2}$, and the claim follows. $\Box$

I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more specifically, the arithmetic regularity and counting lemmas from this paper of Ben Green and myself) to obtain a slight amount of progress towards the open problem of obtaining ${L^p}$ bounds for the trilinear and higher Hilbert transforms (as discussed in this previous blog post). For instance, the trilinear Hilbert transform

$\displaystyle H_3( f_1, f_2, f_3 )(x) := p.v. \int_{\bf R} f_1(x+t) f_2(x+2t) f_3(x+3t)\ \frac{dt}{t}$

is not known to be bounded for any ${L^{p_1}({\bf R}) \times L^{p_2}({\bf R}) \times L^{p_3}({\bf R})}$ to ${L^p({\bf R})}$, although it is conjectured to do so when ${1/p =1/p_1 +1/p_2+1/p_3}$ and ${1 < p_1,p_2,p_3,p < \infty}$. (For ${p}$ well below ${1}$, one can use additive combinatorics constructions to demonstrate unboundedness; see this paper of Demeter.) One can approach this problem by considering the truncated trilinear Hilbert transforms

$\displaystyle H_{3,r,R}( f_1, f_2, f_3 )(x) := \int_{r \leq |t| \leq R} f_1(x+t) f_2(x+2t) f_3(x+3t)\ \frac{dt}{t}$

for ${0 < r < R}$. It is not difficult to show that the boundedness of ${H_3}$ is equivalent to the boundedness of ${H_{3,r,R}}$ with bounds that are uniform in ${R}$ and ${r}$. On the other hand, from Minkowski’s inequality and Hölder’s inequality one can easily obtain the non-uniform bound of ${2 \log \frac{R}{r}}$ for ${H_{3,r,R}}$. The main result of this paper is a slight improvement of this trivial bound to ${o( \log \frac{R}{r})}$ as ${R/r \rightarrow \infty}$. Roughly speaking, the way this gain is established is as follows. First there are some standard time-frequency type reductions to reduce to the task of obtaining some non-trivial cancellation on a single “tree”. Using a “generalised von Neumann theorem”, we show that such cancellation will happen if (a discretised version of) one or more of the functions ${f_1,f_2,f_3}$ (or a dual function ${f_0}$ that it is convenient to test against) is small in the Gowers ${U^3}$ norm. However, the arithmetic regularity lemma alluded to earlier allows one to represent an arbitrary function ${f_i}$, up to a small error, as the sum of such a “Gowers uniform” function, plus a structured function (or more precisely, an irrational virtual nilsequence). This effectively reduces the problem to that of establishing some cancellation in a single tree in the case when all functions ${f_0,f_1,f_2,f_3}$ involved are irrational virtual nilsequences. At this point, the contribution of each component of the tree can be estimated using the “counting lemma” from my paper with Ben. The main term in the asymptotics is a certain integral over a nilmanifold, but because the kernel ${\frac{dt}{t}}$ in the trilinear Hilbert transform is odd, it turns out that this integral vanishes, giving the required cancellation.

The same argument works for higher order Hilbert transforms (and one can also replace the coefficients in these transforms with other rational constants). However, because the quantitative bounds in the arithmetic regularity and counting lemmas are so poor, it does not seem likely that one can use these methods to remove the logarithmic growth in ${R/r}$ entirely, and some additional ideas will be needed to resolve the full conjecture.

In 1946, Ulam, in response to a theorem of Anning and Erdös, posed the following problem:

Problem 1 (Erdös-Ulam problem) Let ${S \subset {\bf R}^2}$ be a set such that the distance between any two points in ${S}$ is rational. Is it true that ${S}$ cannot be (topologically) dense in ${{\bf R}^2}$?

The paper of Anning and Erdös addressed the case that all the distances between two points in ${S}$ were integer rather than rational in the affirmative.

The Erdös-Ulam problem remains open; it was discussed recently over at Gödel’s lost letter. It is in fact likely (as we shall see below) that the set ${S}$ in the above problem is not only forbidden to be topologically dense, but also cannot be Zariski dense either. If so, then the structure of ${S}$ is quite restricted; it was shown by Solymosi and de Zeeuw that if ${S}$ fails to be Zariski dense, then all but finitely many of the points of ${S}$ must lie on a single line, or a single circle. (Conversely, it is easy to construct examples of dense subsets of a line or circle in which all distances are rational, though in the latter case the square of the radius of the circle must also be rational.)

The main tool of the Solymosi-de Zeeuw analysis was Faltings’ celebrated theorem that every algebraic curve of genus at least two contains only finitely many rational points. The purpose of this post is to observe that an affirmative answer to the full Erdös-Ulam problem similarly follows from the conjectured analogue of Falting’s theorem for surfaces, namely the following conjecture of Bombieri and Lang:

Conjecture 2 (Bombieri-Lang conjecture) Let ${X}$ be a smooth projective irreducible algebraic surface defined over the rationals ${{\bf Q}}$ which is of general type. Then the set ${X({\bf Q})}$ of rational points of ${X}$ is not Zariski dense in ${X}$.

In fact, the Bombieri-Lang conjecture has been made for varieties of arbitrary dimension, and for more general number fields than the rationals, but the above special case of the conjecture is the only one needed for this application. We will review what “general type” means (for smooth projective complex varieties, at least) below the fold.

The Bombieri-Lang conjecture is considered to be extremely difficult, in particular being substantially harder than Faltings’ theorem, which is itself a highly non-trivial result. So this implication should not be viewed as a practical route to resolving the Erdös-Ulam problem unconditionally; rather, it is a demonstration of the power of the Bombieri-Lang conjecture. Still, it was an instructive algebraic geometry exercise for me to carry out the details of this implication, which quickly boils down to verifying that a certain quite explicit algebraic surface is of general type (Theorem 4 below). As I am not an expert in the subject, my computations here will be rather tedious and pedestrian; it is likely that they could be made much slicker by exploiting more of the machinery of modern algebraic geometry, and I would welcome any such streamlining by actual experts in this area. (For similar reasons, there may be more typos and errors than usual in this post; corrections are welcome as always.) My calculations here are based on a similar calculation of van Luijk, who used analogous arguments to show (assuming Bombieri-Lang) that the set of perfect cuboids is not Zariski-dense in its projective parameter space.

We also remark that in a recent paper of Makhul and Shaffaf, the Bombieri-Lang conjecture (or more precisely, a weaker consequence of that conjecture) was used to show that if ${S}$ is a subset of ${{\bf R}^2}$ with rational distances which intersects any line in only finitely many points, then there is a uniform bound on the cardinality of the intersection of ${S}$ with any line. I have also recently learned (private communication) that an unpublished work of Shaffaf has obtained a result similar to the one in this post, namely that the Erdös-Ulam conjecture follows from the Bombieri-Lang conjecture, plus an additional conjecture about the rational curves in a specific surface.

Let us now give the elementary reductions to the claim that a certain variety is of general type. For sake of contradiction, let ${S}$ be a dense set such that the distance between any two points is rational. Then ${S}$ certainly contains two points that are a rational distance apart. By applying a translation, rotation, and a (rational) dilation, we may assume that these two points are ${(0,0)}$ and ${(1,0)}$. As ${S}$ is dense, there is a third point of ${S}$ not on the ${x}$ axis, which after a reflection we can place in the upper half-plane; we will write it as ${(a,\sqrt{b})}$ with ${b>0}$.

Given any two points ${P, Q}$ in ${S}$, the quantities ${|P|^2, |Q|^2, |P-Q|^2}$ are rational, and so by the cosine rule the dot product ${P \cdot Q}$ is rational as well. Since ${(1,0) \in S}$, this implies that the ${x}$-component of every point ${P}$ in ${S}$ is rational; this in turn implies that the product of the ${y}$-coordinates of any two points ${P,Q}$ in ${S}$ is rational as well (since this differs from ${P \cdot Q}$ by a rational number). In particular, ${a}$ and ${b}$ are rational, and all of the points in ${S}$ now lie in the lattice ${\{ ( x, y\sqrt{b}): x, y \in {\bf Q} \}}$. (This fact appears to have first been observed in the 1988 habilitationschrift of Kemnitz.)

Now take four points ${(x_j,y_j \sqrt{b})}$, ${j=1,\dots,4}$ in ${S}$ in general position (so that the octuplet ${(x_1,y_1\sqrt{b},\dots,x_4,y_4\sqrt{b})}$ avoids any pre-specified hypersurface in ${{\bf C}^8}$); this can be done if ${S}$ is dense. (If one wished, one could re-use the three previous points ${(0,0), (1,0), (a,\sqrt{b})}$ to be three of these four points, although this ultimately makes little difference to the analysis.) If ${(x,y\sqrt{b})}$ is any point in ${S}$, then the distances ${r_j}$ from ${(x,y\sqrt{b})}$ to ${(x_j,y_j\sqrt{b})}$ are rationals that obey the equations

$\displaystyle (x - x_j)^2 + b (y-y_j)^2 = r_j^2$

for ${j=1,\dots,4}$, and thus determine a rational point in the affine complex variety ${V = V_{b,x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4} \subset {\bf C}^5}$ defined as

$\displaystyle V := \{ (x,y,r_1,r_2,r_3,r_4) \in {\bf C}^6:$

$\displaystyle (x - x_j)^2 + b (y-y_j)^2 = r_j^2 \hbox{ for } j=1,\dots,4 \}.$

By inspecting the projection ${(x,y,r_1,r_2,r_3,r_4) \rightarrow (x,y)}$ from ${V}$ to ${{\bf C}^2}$, we see that ${V}$ is a branched cover of ${{\bf C}^2}$, with the generic cover having ${2^4=16}$ points (coming from the different ways to form the square roots ${r_1,r_2,r_3,r_4}$); in particular, ${V}$ is a complex affine algebraic surface, defined over the rationals. By inspecting the monodromy around the four singular base points ${(x,y) = (x_i,y_i)}$ (which switch the sign of one of the roots ${r_i}$, while keeping the other three roots unchanged), we see that the variety ${V}$ is connected away from its singular set, and thus irreducible. As ${S}$ is topologically dense in ${{\bf R}^2}$, it is Zariski-dense in ${{\bf C}^2}$, and so ${S}$ generates a Zariski-dense set of rational points in ${V}$. To solve the Erdös-Ulam problem, it thus suffices to show that

Claim 3 For any non-zero rational ${b}$ and for rationals ${x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4}$ in general position, the rational points of the affine surface ${V = V_{b,x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4}}$ is not Zariski dense in ${V}$.

This is already very close to a claim that can be directly resolved by the Bombieri-Lang conjecture, but ${V}$ is affine rather than projective, and also contains some singularities. The first issue is easy to deal with, by working with the projectivisation

$\displaystyle \overline{V} := \{ [X,Y,Z,R_1,R_2,R_3,R_4] \in {\bf CP}^6: Q(X,Y,Z,R_1,R_2,R_3,R_4) = 0 \} \ \ \ \ \ (1)$

of ${V}$, where ${Q: {\bf C}^7 \rightarrow {\bf C}^4}$ is the homogeneous quadratic polynomial

$\displaystyle (X,Y,Z,R_1,R_2,R_3,R_4) := (Q_j(X,Y,Z,R_1,R_2,R_3,R_4) )_{j=1}^4$

with

$\displaystyle Q_j(X,Y,Z,R_1,R_2,R_3,R_4) := (X-x_j Z)^2 + b (Y-y_jZ)^2 - R_j^2$

and the projective complex space ${{\bf CP}^6}$ is the space of all equivalence classes ${[X,Y,Z,R_1,R_2,R_3,R_4]}$ of tuples ${(X,Y,Z,R_1,R_2,R_3,R_4) \in {\bf C}^7 \backslash \{0\}}$ up to projective equivalence ${(\lambda X, \lambda Y, \lambda Z, \lambda R_1, \lambda R_2, \lambda R_3, \lambda R_4) \sim (X,Y,Z,R_1,R_2,R_3,R_4)}$. By identifying the affine point ${(x,y,r_1,r_2,r_3,r_4)}$ with the projective point ${(X,Y,1,R_1,R_2,R_3,R_4)}$, we see that ${\overline{V}}$ consists of the affine variety ${V}$ together with the set ${\{ [X,Y,0,R_1,R_2,R_3,R_4]: X^2+bY^2=R^2; R_j = \pm R_1 \hbox{ for } j=2,3,4\}}$, which is the union of eight curves, each of which lies in the closure of ${V}$. Thus ${\overline{V}}$ is the projective closure of ${V}$, and is thus a complex irreducible projective surface, defined over the rationals. As ${\overline{V}}$ is cut out by four quadric equations in ${{\bf CP}^6}$ and has degree sixteen (as can be seen for instance by inspecting the intersection of ${\overline{V}}$ with a generic perturbation of a fibre over the generically defined projection ${[X,Y,Z,R_1,R_2,R_3,R_4] \mapsto [X,Y,Z]}$), it is also a complete intersection. To show (3), it then suffices to show that the rational points in ${\overline{V}}$ are not Zariski dense in ${\overline{V}}$.

Heuristically, the reason why we expect few rational points in ${\overline{V}}$ is as follows. First observe from the projective nature of (1) that every rational point is equivalent to an integer point. But for a septuple ${(X,Y,Z,R_1,R_2,R_3,R_4)}$ of integers of size ${O(N)}$, the quantity ${Q(X,Y,Z,R_1,R_2,R_3,R_4)}$ is an integer point of ${{\bf Z}^4}$ of size ${O(N^2)}$, and so should only vanish about ${O(N^{-8})}$ of the time. Hence the number of integer points ${(X,Y,Z,R_1,R_2,R_3,R_4) \in {\bf Z}^7}$ of height comparable to ${N}$ should be about

$\displaystyle O(N)^7 \times O(N^{-8}) = O(N^{-1});$

this is a convergent sum if ${N}$ ranges over (say) powers of two, and so from standard probabilistic heuristics (see this previous post) we in fact expect only finitely many solutions, in the absence of any special algebraic structure (e.g. the structure of an abelian variety, or a birational reduction to a simpler variety) that could produce an unusually large number of solutions.

The Bombieri-Lang conjecture, Conjecture 2, can be viewed as a formalisation of the above heuristics (roughly speaking, it is one of the most optimistic natural conjectures one could make that is compatible with these heuristics while also being invariant under birational equivalence).

Unfortunately, ${\overline{V}}$ contains some singular points. Being a complete intersection, this occurs when the Jacobian matrix of the map ${Q: {\bf C}^7 \rightarrow {\bf C}^4}$ has less than full rank, or equivalently that the gradient vectors

$\displaystyle \nabla Q_j = (2(X-x_j Z), 2(Y-y_j Z), -2x_j (X-x_j Z) - 2y_j (Y-y_j Z), \ \ \ \ \ (2)$

$\displaystyle 0, \dots, 0, -2R_j, 0, \dots, 0)$

for ${j=1,\dots,4}$ are linearly dependent, where the ${-2R_j}$ is in the coordinate position associated to ${R_j}$. One way in which this can occur is if one of the gradient vectors ${\nabla Q_j}$ vanish identically. This occurs at precisely ${4 \times 2^3 = 32}$ points, when ${[X,Y,Z]}$ is equal to ${[x_j,y_j,1]}$ for some ${j=1,\dots,4}$, and one has ${R_k = \pm ( (x_j - x_k)^2 + b (y_j - y_k)^2 )^{1/2}}$ for all ${k=1,\dots,4}$ (so in particular ${R_j=0}$). Let us refer to these as the obvious singularities; they arise from the geometrically evident fact that the distance function ${(x,y\sqrt{b}) \mapsto \sqrt{(x-x_j)^2 + b(y-y_j)^2}}$ is singular at ${(x_j,y_j\sqrt{b})}$.

The other way in which could occur is if a non-trivial linear combination of at least two of the gradient vectors vanishes. From (2), this can only occur if ${R_j=R_k=0}$ for some distinct ${j,k}$, which from (1) implies that

$\displaystyle (X - x_j Z) = \pm \sqrt{b} i (Y - y_j Z) \ \ \ \ \ (3)$

and

$\displaystyle (X - x_k Z) = \pm \sqrt{b} i (Y - y_k Z) \ \ \ \ \ (4)$

for two choices of sign ${\pm}$. If the signs are equal, then (as ${x_j, y_j, x_k, y_k}$ are in general position) this implies that ${Z=0}$, and then we have the singular point

$\displaystyle [X,Y,Z,R_1,R_2,R_3,R_4] = [\pm \sqrt{b} i, 1, 0, 0, 0, 0, 0]. \ \ \ \ \ (5)$

If the non-trivial linear combination involved three or more gradient vectors, then by the pigeonhole principle at least two of the signs involved must be equal, and so the only singular points are (5). So the only remaining possibility is when we have two gradient vectors ${\nabla Q_j, \nabla Q_k}$ that are parallel but non-zero, with the signs in (3), (4) opposing. But then (as ${x_j,y_j,x_k,y_k}$ are in general position) the vectors ${(X-x_j Z, Y-y_j Z), (X-x_k Z, Y-y_k Z)}$ are non-zero and non-parallel to each other, a contradiction. Thus, outside of the ${32}$ obvious singular points mentioned earlier, the only other singular points are the two points (5).

We will shortly show that the ${32}$ obvious singularities are ordinary double points; the surface ${\overline{V}}$ near any of these points is analytically equivalent to an ordinary cone ${\{ (x,y,z) \in {\bf C}^3: z^2 = x^2 + y^2 \}}$ near the origin, which is a cone over a smooth conic curve ${\{ (x,y) \in {\bf C}^2: x^2+y^2=1\}}$. The two non-obvious singularities (5) are slightly more complicated than ordinary double points, they are elliptic singularities, which approximately resemble a cone over an elliptic curve. (As far as I can tell, this resemblance is exact in the category of real smooth manifolds, but not in the category of algebraic varieties.) If one blows up each of the point singularities of ${\overline{V}}$ separately, no further singularities are created, and one obtains a smooth projective surface ${X}$ (using the Segre embedding as necessary to embed ${X}$ back into projective space, rather than in a product of projective spaces). Away from the singularities, the rational points of ${\overline{V}}$ lift up to rational points of ${X}$. Assuming the Bombieri-Lang conjecture, we thus are able to answer the Erdös-Ulam problem in the affirmative once we establish

Theorem 4 The blowup ${X}$ of ${\overline{V}}$ is of general type.

This will be done below the fold, by the pedestrian device of explicitly constructing global differential forms on ${X}$; I will also be working from a complex analysis viewpoint rather than an algebraic geometry viewpoint as I am more comfortable with the former approach. (As mentioned above, though, there may well be a quicker way to establish this result by using more sophisticated machinery.)

I thank Mark Green and David Gieseker for helpful conversations (and a crash course in varieties of general type!).

Remark 5 The above argument shows in fact (assuming Bombieri-Lang) that sets ${S \subset {\bf R}^2}$ with all distances rational cannot be Zariski-dense, and thus (by Solymosi-de Zeeuw) must lie on a single line or circle with only finitely many exceptions. Assuming a stronger version of Bombieri-Lang involving a general number field ${K}$, we obtain a similar conclusion with “rational” replaced by “lying in ${K}$” (one has to extend the Solymosi-de Zeeuw analysis to more general number fields, but this should be routine, using the analogue of Faltings’ theorem for such number fields).

Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple spectrum“. Recall that an ${n \times n}$ Hermitian matrix is said to have simple eigenvalues if all of its ${n}$ eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all ${n \times n}$ Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we conclude that random matrices drawn from this ensemble will almost surely have simple eigenvalues.

For discrete random matrix ensembles, though, the above argument breaks down, even though general universality heuristics predict that the statistics of discrete ensembles should behave similarly to those of continuous ensembles. A model case here is the adjacency matrix ${M_n}$ of an Erdös-Rényi graph – a graph on ${n}$ vertices in which any pair of vertices has an independent probability ${p}$ of being in the graph. For the purposes of this paper one should view ${p}$ as fixed, e.g. ${p=1/2}$, while ${n}$ is an asymptotic parameter going to infinity. In this context, our main result is the following (answering a question of Babai):

Theorem 1 With probability ${1-o(1)}$, ${M_n}$ has simple eigenvalues.

Our argument works for more general Wigner-type matrix ensembles, but for sake of illustration we will stick with the Erdös-Renyi case. Previous work on local universality for such matrix models (e.g. the work of Erdos, Knowles, Yau, and Yin) was able to show that any individual eigenvalue gap ${\lambda_{i+1}(M)-\lambda_i(M)}$ did not vanish with probability ${1-o(1)}$ (in fact ${1-O(n^{-c})}$ for some absolute constant ${c>0}$), but because there are ${n}$ different gaps that one has to simultaneously ensure to be non-zero, this did not give Theorem 1 as one is forced to apply the union bound.

Our argument in fact gives simplicity of the spectrum with probability ${1-O(n^{-A})}$ for any fixed ${A}$; in a subsequent paper we also show that it gives a quantitative lower bound on the eigenvalue gaps (analogous to how many results on the singularity probability of random matrices can be upgraded to a bound on the least singular value).

The basic idea of argument can be sketched as follows. Suppose that ${M_n}$ has a repeated eigenvalue ${\lambda}$. We split

$\displaystyle M_n = \begin{pmatrix} M_{n-1} & X \\ X^T & 0 \end{pmatrix}$

for a random ${n-1 \times n-1}$ minor ${M_{n-1}}$ and a random sign vector ${X}$; crucially, ${X}$ and ${M_{n-1}}$ are independent. If ${M_n}$ has a repeated eigenvalue ${\lambda}$, then by the Cauchy interlacing law, ${M_{n-1}}$ also has an eigenvalue ${\lambda}$. We now write down the eigenvector equation for ${M_n}$ at ${\lambda}$:

$\displaystyle \begin{pmatrix} M_{n-1} & X \\ X^T & 0 \end{pmatrix} \begin{pmatrix} v \\ a \end{pmatrix} = \lambda \begin{pmatrix} v \\ a \end{pmatrix}.$

Extracting the top ${n-1}$ coefficients, we obtain

$\displaystyle (M_{n-1} - \lambda) v + a X = 0.$

If we let ${w}$ be the ${\lambda}$-eigenvector of ${M_{n-1}}$, then by taking inner products with ${w}$ we conclude that

$\displaystyle a (w \cdot X) = 0;$

we typically expect ${a}$ to be non-zero, in which case we arrive at

$\displaystyle w \cdot X = 0.$

In other words, in order for ${M_n}$ to have a repeated eigenvalue, the top right column ${X}$ of ${M_n}$ has to be orthogonal to an eigenvector ${w}$ of the minor ${M_{n-1}}$. Note that ${X}$ and ${w}$ are going to be independent (once we specify which eigenvector of ${M_{n-1}}$ to take as ${w}$). On the other hand, thanks to inverse Littlewood-Offord theory (specifically, we use an inverse Littlewood-Offord theorem of Nguyen and Vu), we know that the vector ${X}$ is unlikely to be orthogonal to any given vector ${w}$ independent of ${X}$, unless the coefficients of ${w}$ are extremely special (specifically, that most of them lie in a generalised arithmetic progression). The main remaining difficulty is then to show that eigenvectors of a random matrix are typically not of this special form, and this relies on a conditioning argument originally used by Komlós to bound the singularity probability of a random sign matrix. (Basically, if an eigenvector has this special form, then one can use a fraction of the rows and columns of the random matrix to determine the eigenvector completely, while still preserving enough randomness in the remaining portion of the matrix so that this vector will in fact not be an eigenvector with high probability.)

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.

In addition to the Fields medallists mentioned in the previous post, the IMU also awarded the Nevanlinna prize to Subhash Khot, the Gauss prize to Stan Osher (my colleague here at UCLA!), and the Chern medal to Phillip Griffiths. Like I did in 2010, I’ll try to briefly discuss one result of each of the prize winners, though the fields of mathematics here are even further from my expertise than those discussed in the previous post (and all the caveats from that post apply here also).

Subhash Khot is best known for his Unique Games Conjecture, a problem in complexity theory that is perhaps second in importance only to the ${P \neq NP}$ problem for the purposes of demarcating the mysterious line between “easy” and “hard” problems (if one follows standard practice and uses “polynomial time” as the definition of “easy”). The ${P \neq NP}$ problem can be viewed as an assertion that it is difficult to find exact solutions to certain standard theoretical computer science problems (such as ${k}$-SAT); thanks to the NP-completeness phenomenon, it turns out that the precise problem posed here is not of critical importance, and ${k}$-SAT may be substituted with one of the many other problems known to be NP-complete. The unique games conjecture is similarly an assertion about the difficulty of finding even approximate solutions to certain standard problems, in particular “unique games” problems in which one needs to colour the vertices of a graph in such a way that the colour of one vertex of an edge is determined uniquely (via a specified matching) by the colour of the other vertex. This is an easy problem to solve if one insists on exact solutions (in which 100% of the edges have a colouring compatible with the specified matching), but becomes extremely difficult if one permits approximate solutions, with no exact solution available. In analogy with the NP-completeness phenomenon, the threshold for approximate satisfiability of many other problems (such as the MAX-CUT problem) is closely connected with the truth of the unique games conjecture; remarkably, the truth of the unique games conjecture would imply asymptotically sharp thresholds for many of these problems. This has implications for many theoretical computer science constructions which rely on hardness of approximation, such as probabilistically checkable proofs. For a more detailed survey of the unique games conjecture and its implications, see this Bulletin article of Trevisan.

My colleague Stan Osher has worked in many areas of applied mathematics, ranging from image processing to modeling fluids for major animation studies such as Pixar or Dreamworks, but today I would like to talk about one of his contributions that is close to an area of my own expertise, namely compressed sensing. One of the basic reconstruction problem in compressed sensing is the basis pursuit problem of finding the vector ${x \in {\bf R}^n}$ in an affine space ${\{ x \in {\bf R}^n: Ax = b \}}$ (where ${b \in {\bf R}^m}$ and ${A \in {\bf R}^{m\times n}}$ are given, and ${m}$ is typically somewhat smaller than ${n}$) which minimises the ${\ell^1}$-norm ${\|x\|_{\ell^1} := \sum_{i=1}^n |x_i|}$ of the vector ${x}$. This is a convex optimisation problem, and thus solvable in principle (it is a polynomial time problem, and thus “easy” in the above theoretical computer science sense). However, once ${n}$ and ${m}$ get moderately large (e.g. of the order of ${10^6}$), standard linear optimisation routines begin to become computationally expensive; also, it is difficult for off-the-shelf methods to exploit any additional structure (e.g. sparsity) in the measurement matrix ${A}$. Much of the problem comes from the fact that the functional ${x \mapsto \|x\|_1}$ is only barely convex. One way to speed up the optimisation problem is to relax it by replacing the constraint ${Ax=b}$ with a convex penalty term ${\frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2}$, thus one is now trying to minimise the unconstrained functional

$\displaystyle \|x\|_1 + \frac{1}{2\epsilon} \|Ax-b\|_{\ell^2}^2.$

This functional is more convex, and is over a computationally simpler domain ${{\bf R}^n}$ than the affine space ${\{x \in {\bf R}^n: Ax=b\}}$, so is easier (though still not entirely trivial) to optimise over. However, the minimiser ${x^\epsilon}$ to this problem need not match the minimiser ${x^0}$ to the original problem, particularly if the (sub-)gradient ${\partial \|x\|_1}$ of the original functional ${\|x\|_1}$ is large at ${x^0}$, and if ${\epsilon}$ is not set to be small. (And setting ${\epsilon}$ too small will cause other difficulties with numerically solving the optimisation problem, due to the need to divide by very small denominators.) However, if one modifies the objective function by an additional linear term

$\displaystyle \|x\|_1 - \langle p, x \rangle + \frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2$

then some simple convexity considerations reveal that the minimiser to this new problem will match the minimiser ${x^0}$ to the original problem, provided that ${p}$ is (or more precisely, lies in) the (sub-)gradient ${\partial \|x\|_1}$ of ${\|x\|_1}$ at ${x^0}$ – even if ${\epsilon}$ is not small. But, one does not know in advance what the correct value of ${p}$ should be, because one does not know what the minimiser ${x^0}$ is.

With Yin, Goldfarb and Darbon, Osher introduced a Bregman iteration method in which one solves for ${x}$ and ${p}$ simultaneously; given an initial guess ${x^k, p^k}$ for both ${x^k}$ and ${p^k}$, one first updates ${x^k}$ to the minimiser ${x^{k+1} \in {\bf R}^n}$ of the convex functional

$\displaystyle \|x\|_1 - \langle p^k, x \rangle + \frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2 \ \ \ \ \ (1)$

and then updates ${p^{k+1}}$ to the natural value of the subgradient ${\partial \|x\|_1}$ at ${x^{k+1}}$, namely

$\displaystyle p^{k+1} := p^k - \nabla \frac{1}{2 \epsilon} \|Ax-b\|_{\ell^2}^2|_{x=x^{k+1}} = p_k - \frac{1}{\epsilon} (Ax^k - b)$

(note upon taking the first variation of (1) that ${p^{k+1}}$ is indeed in the subgradient). This procedure converges remarkably quickly (both in theory and in practice) to the true minimiser ${x^0}$ even for non-small values of ${\epsilon}$, and also has some ability to be parallelised, and has led to actual performance improvements of an order of magnitude or more in certain compressed sensing problems (such as reconstructing an MRI image).

Phillip Griffiths has made many contributions to complex, algebraic and differential geometry, and I am not qualified to describe most of these; my primary exposure to his work is through his text on algebraic geometry with Harris, but as excellent though that text is, it is not really representative of his research. But I thought I would mention one cute result of his related to the famous Nash embedding theorem. Suppose that one has a smooth ${n}$-dimensional Riemannian manifold that one wants to embed locally into a Euclidean space ${{\bf R}^m}$. The Nash embedding theorem guarantees that one can do this if ${m}$ is large enough depending on ${n}$, but what is the minimal value of ${m}$ one can get away with? Many years ago, my colleague Robert Greene showed that ${m = \frac{n(n+1)}{2} + n}$ sufficed (a simplified proof was subsequently given by Gunther). However, this is not believed to be sharp; if one replaces “smooth” with “real analytic” then a standard Cauchy-Kovalevski argument shows that ${m = \frac{n(n+1)}{2}}$ is possible, and no better. So this suggests that ${m = \frac{n(n+1)}{2}}$ is the threshold for the smooth problem also, but this remains open in general. The cases ${n=1}$ is trivial, and the ${n=2}$ case is not too difficult (if the curvature is non-zero) as the codimension ${m-n}$ is one in this case, and the problem reduces to that of solving a Monge-Ampere equation. With Bryant and Yang, Griffiths settled the ${n=3}$ case, under a non-degeneracy condition on the Einstein tensor. This is quite a serious paper – over 100 pages combining differential geometry, PDE methods (e.g. Nash-Moser iteration), and even some harmonic analysis (e.g. they rely at one key juncture on an extension theorem of my advisor, Elias Stein). The main difficulty is that that the relevant PDE degenerates along a certain characteristic submanifold of the cotangent bundle, which then requires an extremely delicate analysis to handle.

This is a blog version of a talk I recently gave at the IPAM workshop on “The Kakeya Problem, Restriction Problem, and Sum-product Theory”.

Note: the discussion here will be highly non-rigorous in nature, being extremely loose in particular with asymptotic notation and with the notion of dimension. Caveat emptor.

One of the most infamous unsolved problems at the intersection of geometric measure theory, incidence combinatorics, and real-variable harmonic analysis is the Kakeya set conjecture. We will focus on the following three-dimensional case of the conjecture, stated informally as follows:

Conjecture 1 (Kakeya conjecture) Let ${E}$ be a subset of ${{\bf R}^3}$ that contains a unit line segment in every direction. Then ${\hbox{dim}(E) = 3}$.

This conjecture is not precisely formulated here, because we have not specified exactly what type of set ${E}$ is (e.g. measurable, Borel, compact, etc.) and what notion of dimension we are using. We will deliberately ignore these technical details in this post. It is slightly more convenient for us here to work with lines instead of unit line segments, so we work with the following slight variant of the conjecture (which is essentially equivalent):

Conjecture 2 (Kakeya conjecture, again) Let ${{\cal L}}$ be a family of lines in ${{\bf R}^3}$ that meet ${B(0,1)}$ and contain a line in each direction. Let ${E}$ be the union of the restriction ${\ell \cap B(0,2)}$ to ${B(0,2)}$ of every line ${\ell}$ in ${{\cal L}}$. Then ${\hbox{dim}(E) = 3}$.

As the space of all directions in ${{\bf R}^3}$ is two-dimensional, we thus see that ${{\cal L}}$ is an (at least) two-dimensional subset of the four-dimensional space of lines in ${{\bf R}^3}$ (actually, it lies in a compact subset of this space, since we have constrained the lines to meet ${B(0,1)}$). One could then ask if this is the only property of ${{\cal L}}$ that is needed to establish the Kakeya conjecture, that is to say if any subset of ${B(0,2)}$ which contains a two-dimensional family of lines (restricted to ${B(0,2)}$, and meeting ${B(0,1)}$) is necessarily three-dimensional. Here we have an easy counterexample, namely a plane in ${B(0,2)}$ (passing through the origin), which contains a two-dimensional collection of lines. However, we can exclude this case by adding an additional axiom, leading to what one might call a “strong” Kakeya conjecture:

Conjecture 3 (Strong Kakeya conjecture) Let ${{\cal L}}$ be a two-dimensional family of lines in ${{\bf R}^3}$ that meet ${B(0,1)}$, and assume the Wolff axiom that no (affine) plane contains more than a one-dimensional family of lines in ${{\cal L}}$. Let ${E}$ be the union of the restriction ${\ell \cap B(0,2)}$ of every line ${\ell}$ in ${{\cal L}}$. Then ${\hbox{dim}(E) = 3}$.

Actually, to make things work out we need a more quantitative version of the Wolff axiom in which we constrain the metric entropy (and not just dimension) of lines that lie close to a plane, rather than exactly on the plane. However, for the informal discussion here we will ignore these technical details. Families of lines that lie in different directions will obey the Wolff axiom, but the converse is not true in general.

In 1995, Wolff established the important lower bound ${\hbox{dim}(E) \geq 5/2}$ (for various notions of dimension, e.g. Hausdorff dimension) for sets ${E}$ in Conjecture 3 (and hence also for the other forms of the Kakeya problem). However, there is a key obstruction to going beyond the ${5/2}$ barrier, coming from the possible existence of half-dimensional (approximate) subfields of the reals ${{\bf R}}$. To explain this problem, it easiest to first discuss the complex version of the strong Kakeya conjecture, in which all relevant (real) dimensions are doubled:

Conjecture 4 (Strong Kakeya conjecture over ${{\bf C}}$) Let ${{\cal L}}$ be a four (real) dimensional family of complex lines in ${{\bf C}^3}$ that meet the unit ball ${B(0,1)}$ in ${{\bf C}^3}$, and assume the Wolff axiom that no four (real) dimensional (affine) subspace contains more than a two (real) dimensional family of complex lines in ${{\cal L}}$. Let ${E}$ be the union of the restriction ${\ell \cap B(0,2)}$ of every complex line ${\ell}$ in ${{\cal L}}$. Then ${E}$ has real dimension ${6}$.

The argument of Wolff can be adapted to the complex case to show that all sets ${E}$ occuring in Conjecture 4 have real dimension at least ${5}$. Unfortunately, this is sharp, due to the following fundamental counterexample:

Proposition 5 (Heisenberg group counterexample) Let ${H \subset {\bf C}^3}$ be the Heisenberg group

$\displaystyle H = \{ (z_1,z_2,z_3) \in {\bf C}^3: \hbox{Im}(z_1) = \hbox{Im}(z_2 \overline{z_3}) \}$

and let ${{\cal L}}$ be the family of complex lines

$\displaystyle \ell_{s,t,\alpha} := \{ (\overline{\alpha} z + t, z, sz + \alpha): z \in {\bf C} \}$

with ${s,t \in {\bf R}}$ and ${\alpha \in {\bf C}}$. Then ${H}$ is a five (real) dimensional subset of ${{\bf C}^3}$ that contains every line in the four (real) dimensional set ${{\cal L}}$; however each four real dimensional (affine) subspace contains at most a two (real) dimensional set of lines in ${{\cal L}}$. In particular, the strong Kakeya conjecture over the complex numbers is false.

This proposition is proven by a routine computation, which we omit here. The group structure on ${H}$ is given by the group law

$\displaystyle (z_1,z_2,z_3) \cdot (w_1,w_2,w_3) = (z_1 + w_1 + z_2 \overline{w_3} - z_3 \overline{w_2}, z_2 +w_2, z_3+w_3),$

giving ${E}$ the structure of a ${2}$-step simply-connected nilpotent Lie group, isomorphic to the usual Heisenberg group over ${{\bf R}^2}$. Note that while the Heisenberg group is a counterexample to the complex strong Kakeya conjecture, it is not a counterexample to the complex form of the original Kakeya conjecture, because the complex lines ${{\cal L}}$ in the Heisenberg counterexample do not point in distinct directions, but instead only point in a three (real) dimensional subset of the four (real) dimensional space of available directions for complex lines. For instance, one has the one real-dimensional family of parallel lines

$\displaystyle \ell_{0,t,0} = \{ (t, z, 0): z \in {\bf C}\}$

with ${t \in {\bf R}}$; multiplying this family of lines on the right by a group element in ${H}$ gives other families of parallel lines, which in fact sweep out all of ${{\cal L}}$.

The Heisenberg counterexample ultimately arises from the “half-dimensional” (and hence degree two) subfield ${{\bf R}}$ of ${{\bf C}}$, which induces an involution ${z \mapsto \overline{z}}$ which can then be used to define the Heisenberg group ${H}$ through the formula

$\displaystyle H = \{ (z_1,z_2,z_3) \in {\bf C}^3: z_1 - \overline{z_1} = z_2 \overline{z_3} - z_3 \overline{z_2} \}.$

Analogous Heisenberg counterexamples can also be constructed if one works over finite fields ${{\bf F}_{q^2}}$ that contain a “half-dimensional” subfield ${{\bf F}_q}$; we leave the details to the interested reader. Morally speaking, if ${{\bf R}}$ in turn contained a subfield of dimension ${1/2}$ (or even a subring or “approximate subring”), then one ought to be able to use this field to generate a counterexample to the strong Kakeya conjecture over the reals. Fortunately, such subfields do not exist; this was a conjecture of Erdos and Volkmann that was proven by Edgar and Miller, and more quantitatively by Bourgain (answering a question of Nets Katz and myself). However, this fact is not entirely trivial to prove, being a key example of the sum-product phenomenon.

We thus see that to go beyond the ${5/2}$ dimension bound of Wolff for the 3D Kakeya problem over the reals, one must do at least one of two things:

• (a) Exploit the distinct directions of the lines in ${{\mathcal L}}$ in a way that goes beyond the Wolff axiom; or
• (b) Exploit the fact that ${{\bf R}}$ does not contain half-dimensional subfields (or more generally, intermediate-dimensional approximate subrings).

(The situation is more complicated in higher dimensions, as there are more obstructions than the Heisenberg group; for instance, in four dimensions quadric surfaces are an important obstruction, as discussed in this paper of mine.)

Various partial or complete results on the Kakeya problem over various fields have been obtained through route (a) or route (b). For instance, in 2000, Nets Katz, Izabella Laba and myself used route (a) to improve Wolff’s lower bound of ${5/2}$ for Kakeya sets very slightly to ${5/2+10^{-10}}$ (for a weak notion of dimension, namely upper Minkowski dimension). In 2004, Bourgain, Katz, and myself established a sum-product estimate which (among other things) ruled out approximate intermediate-dimensional subrings of ${{\bf F}_p}$, and then pursued route (b) to obtain a corresponding improvement ${5/2+\epsilon}$ to the Kakeya conjecture over finite fields of prime order. The analogous (discretised) sum-product estimate over the reals was established by Bourgain in 2003, which in principle would allow one to extend the result of Katz, Laba and myself to the strong Kakeya setting, but this has not been carried out in the literature. Finally, in 2009, Dvir used route (a) and introduced the polynomial method (as discussed previously here) to completely settle the Kakeya conjecture in finite fields.

Below the fold, I present a heuristic argument of Nets Katz and myself, which in principle would use route (b) to establish the full (strong) Kakeya conjecture. In broad terms, the strategy is as follows:

1. Assume that the (strong) Kakeya conjecture fails, so that there are sets ${E}$ of the form in Conjecture 3 of dimension ${3-\sigma}$ for some ${\sigma>0}$. Assume that ${E}$ is “optimal”, in the sense that ${\sigma}$ is as large as possible.
2. Use the optimality of ${E}$ (and suitable non-isotropic rescalings) to establish strong forms of standard structural properties expected of such sets ${E}$, namely “stickiness”, “planiness”, “local graininess” and “global graininess” (we will roughly describe these properties below). Heuristically, these properties are constraining ${E}$ to “behave like” a putative Heisenberg group counterexample.
3. By playing all these structural properties off of each other, show that ${E}$ can be parameterised locally by a one-dimensional set which generates a counterexample to Bourgain’s sum-product theorem. This contradiction establishes the Kakeya conjecture.

Nets and I have had an informal version of argument for many years, but were never able to make a satisfactory theorem (or even a partial Kakeya result) out of it, because we could not rigorously establish anywhere near enough of the necessary structural properties (stickiness, planiness, etc.) on the optimal set ${E}$ for a large number of reasons (one of which being that we did not have a good notion of dimension that did everything that we wished to demand of it). However, there is beginning to be movement in these directions (e.g. in this recent result of Guth using the polynomial method obtaining a weak version of local graininess on certain Kakeya sets). In view of this (and given that neither Nets or I have been actively working in this direction for some time now, due to many other projects), we’ve decided to distribute these ideas more widely than before, and in particular on this blog.

Roth’s theorem on arithmetic progressions asserts that every subset of the integers ${{\bf Z}}$ of positive upper density contains infinitely many arithmetic progressions of length three. There are many versions and variants of this theorem. Here is one of them:

Theorem 1 (Roth’s theorem) Let ${G = (G,+)}$ be a compact abelian group, with Haar probability measure ${\mu}$, which is ${2}$-divisible (i.e. the map ${x \mapsto 2x}$ is surjective) and let ${A}$ be a measurable subset of ${G}$ with ${\mu(A) \geq \alpha}$ for some ${0 < \alpha < 1}$. Then we have

$\displaystyle \int_G \int_G 1_A(x) 1_A(x+r) 1_A(x+2r)\ d\mu(x) d\mu(r) \gg_\alpha 1,$

where ${X \gg_\alpha Y}$ denotes the bound ${X \geq c_\alpha Y}$ for some ${c_\alpha > 0}$ depending only on ${\alpha}$.

This theorem is usually formulated in the case that ${G}$ is a finite abelian group of odd order (in which case the result is essentially due to Meshulam) or more specifically a cyclic group ${G = {\bf Z}/N{\bf Z}}$ of odd order (in which case it is essentially due to Varnavides), but is also valid for the more general setting of ${2}$-divisible compact abelian groups, as we shall shortly see. One can be more precise about the dependence of the implied constant ${c_\alpha}$ on ${\alpha}$, but to keep the exposition simple we will work at the qualitative level here, without trying at all to get good quantitative bounds. The theorem is also true without the ${2}$-divisibility hypothesis, but the proof we will discuss runs into some technical issues due to the degeneracy of the ${2r}$ shift in that case.

We can deduce Theorem 1 from the following more general Khintchine-type statement. Let ${\hat G}$ denote the Pontryagin dual of a compact abelian group ${G}$, that is to say the set of all continuous homomorphisms ${\xi: x \mapsto \xi \cdot x}$ from ${G}$ to the (additive) unit circle ${{\bf R}/{\bf Z}}$. Thus ${\hat G}$ is a discrete abelian group, and functions ${f \in L^2(G)}$ have a Fourier transform ${\hat f \in \ell^2(\hat G)}$ defined by

$\displaystyle \hat f(\xi) := \int_G f(x) e^{-2\pi i \xi \cdot x}\ d\mu(x).$

If ${G}$ is ${2}$-divisible, then ${\hat G}$ is ${2}$-torsion-free in the sense that the map ${\xi \mapsto 2 \xi}$ is injective. For any finite set ${S \subset \hat G}$ and any radius ${\rho>0}$, define the Bohr set

$\displaystyle B(S,\rho) := \{ x \in G: \sup_{\xi \in S} \| \xi \cdot x \|_{{\bf R}/{\bf Z}} < \rho \}$

where ${\|\theta\|_{{\bf R}/{\bf Z}}}$ denotes the distance of ${\theta}$ to the nearest integer. We refer to the cardinality ${|S|}$ of ${S}$ as the rank of the Bohr set. We record a simple volume bound on Bohr sets:

Lemma 2 (Volume packing bound) Let ${G}$ be a compact abelian group with Haar probability measure ${\mu}$. For any Bohr set ${B(S,\rho)}$, we have

$\displaystyle \mu( B( S, \rho ) ) \gg_{|S|, \rho} 1.$

Proof: We can cover the torus ${({\bf R}/{\bf Z})^S}$ by ${O_{|S|,\rho}(1)}$ translates ${\theta+Q}$ of the cube ${Q := \{ (\theta_\xi)_{\xi \in S} \in ({\bf R}/{\bf Z})^S: \sup_{\xi \in S} \|\theta_\xi\|_{{\bf R}/{\bf Z}} < \rho/2 \}}$. Then the sets ${\{ x \in G: (\xi \cdot x)_{\xi \in S} \in \theta + Q \}}$ form an cover of ${G}$. But all of these sets lie in a translate of ${B(S,\rho)}$, and the claim then follows from the translation invariance of ${\mu}$. $\Box$

Given any Bohr set ${B(S,\rho)}$, we define a normalised “Lipschitz” cutoff function ${\nu_{B(S,\rho)}: G \rightarrow {\bf R}}$ by the formula

$\displaystyle \nu_{B(S,\rho)}(x) = c_{B(S,\rho)} (1 - \frac{1}{\rho} \sup_{\xi \in S} \|\xi \cdot x\|_{{\bf R}/{\bf Z}})_+ \ \ \ \ \ (1)$

where ${c_{B(S,\rho)}}$ is the constant such that

$\displaystyle \int_G \nu_{B(S,\rho)}\ d\mu = 1,$

thus

$\displaystyle c_{B(S,\rho)} = \left( \int_{B(S,\rho)} (1 - \frac{1}{\rho} \sup_{\xi \in S} \|\xi \cdot x\|_{{\bf R}/{\bf Z}})\ d\mu(x) \right)^{-1}.$

The function ${\nu_{B(S,\rho)}}$ should be viewed as an ${L^1}$-normalised “tent function” cutoff to ${B(S,\rho)}$. Note from Lemma 2 that

$\displaystyle 1 \ll_{|S|,\rho} c_{B(S,\rho)} \ll_{|S|,\rho} 1. \ \ \ \ \ (2)$

We then have the following sharper version of Theorem 1:

Theorem 3 (Roth-Khintchine theorem) Let ${G = (G,+)}$ be a ${2}$-divisible compact abelian group, with Haar probability measure ${\mu}$, and let ${\epsilon>0}$. Then for any measurable function ${f: G \rightarrow [0,1]}$, there exists a Bohr set ${B(S,\rho)}$ with ${|S| \ll_\epsilon 1}$ and ${\rho \gg_\epsilon 1}$ such that

$\displaystyle \int_G \int_G f(x) f(x+r) f(x+2r) \nu_{B(S,\rho)}*\nu_{B(S,\rho)}(r)\ d\mu(x) d\mu(r) \ \ \ \ \ (3)$

$\displaystyle \geq (\int_G f\ d\mu)^3 - O(\epsilon)$

where ${*}$ denotes the convolution operation

$\displaystyle f*g(x) := \int_G f(y) g(x-y)\ d\mu(y).$

A variant of this result (expressed in the language of ergodic theory) appears in this paper of Bergelson, Host, and Kra; a combinatorial version of the Bergelson-Host-Kra result that is closer to Theorem 3 subsequently appeared in this paper of Ben Green and myself, but this theorem arguably appears implicitly in a much older paper of Bourgain. To see why Theorem 3 implies Theorem 1, we apply the theorem with ${f := 1_A}$ and ${\epsilon}$ equal to a small multiple of ${\alpha^3}$ to conclude that there is a Bohr set ${B(S,\rho)}$ with ${|S| \ll_\alpha 1}$ and ${\rho \gg_\alpha 1}$ such that

$\displaystyle \int_G \int_G 1_A(x) 1_A(x+r) 1_A(x+2r) \nu_{B(S,\rho)}*\nu_{B(S,\rho)}(r)\ d\mu(x) d\mu(r) \gg \alpha^3.$

But from (2) we have the pointwise bound ${\nu_{B(S,\rho)}*\nu_{B(S,\rho)} \ll_\alpha 1}$, and Theorem 1 follows.

Below the fold, we give a short proof of Theorem 3, using an “energy pigeonholing” argument that essentially dates back to the 1986 paper of Bourgain mentioned previously (not to be confused with a later 1999 paper of Bourgain on Roth’s theorem that was highly influential, for instance in emphasising the importance of Bohr sets). The idea is to use the pigeonhole principle to choose the Bohr set ${B(S,\rho)}$ to capture all the “large Fourier coefficients” of ${f}$, but such that a certain “dilate” of ${B(S,\rho)}$ does not capture much more Fourier energy of ${f}$ than ${B(S,\rho)}$ itself. The bound (3) may then be obtained through elementary Fourier analysis, without much need to explicitly compute things like the Fourier transform of an indicator function of a Bohr set. (However, the bound obtained by this argument is going to be quite poor – of tower-exponential type.) To do this we perform a structural decomposition of ${f}$ into “structured”, “small”, and “highly pseudorandom” components, as is common in the subject (e.g. in this previous blog post), but even though we crucially need to retain non-negativity of one of the components in this decomposition, we can avoid recourse to conditional expectation with respect to a partition (or “factor”) of the space, using instead convolution with one of the ${\nu_{B(S,\rho)}}$ considered above to achieve a similar effect.

A core foundation of the subject now known as arithmetic combinatorics (and particularly the subfield of additive combinatorics) are the elementary sum set estimates (sometimes known as “Ruzsa calculus”) that relate the cardinality of various sum sets

$\displaystyle A+B := \{ a+b: a \in A, b \in B \}$

and difference sets

$\displaystyle A-B := \{ a-b: a \in A, b \in B \},$

as well as iterated sumsets such as ${3A=A+A+A}$, ${2A-2A=A+A-A-A}$, and so forth. Here, ${A, B}$ are finite non-empty subsets of some additive group ${G = (G,+)}$ (classically one took ${G={\bf Z}}$ or ${G={\bf R}}$, but nowadays one usually considers more general additive groups). Some basic estimates in this vein are the following:

Lemma 1 (Ruzsa covering lemma) Let ${A, B}$ be finite non-empty subsets of ${G}$. Then ${A}$ may be covered by at most ${\frac{|A+B|}{|B|}}$ translates of ${B-B}$.

Proof: Consider a maximal set of disjoint translates ${a+B}$ of ${B}$ by elements ${a \in A}$. These translates have cardinality ${|B|}$, are disjoint, and lie in ${A+B}$, so there are at most ${\frac{|A+B|}{|B|}}$ of them. By maximality, for any ${a' \in A}$, ${a'+B}$ must intersect at least one of the selected ${a+B}$, thus ${a' \in a+B-B}$, and the claim follows. $\Box$

Lemma 2 (Ruzsa triangle inequality) Let ${A,B,C}$ be finite non-empty subsets of ${G}$. Then ${|A-C| \leq \frac{|A-B| |B-C|}{|B|}}$.

Proof: Consider the addition map ${+: (x,y) \mapsto x+y}$ from ${(A-B) \times (B-C)}$ to ${G}$. Every element ${a-c}$ of ${A - C}$ has a preimage ${\{ (x,y) \in (A-B) \times (B-C)\}}$ of this map of cardinality at least ${|B|}$, thanks to the obvious identity ${a-c = (a-b) + (b-c)}$ for each ${b \in B}$. Since ${(A-B) \times (B-C)}$ has cardinality ${|A-B| |B-C|}$, the claim follows. $\Box$

Such estimates (which are covered, incidentally, in Section 2 of my book with Van Vu) are particularly useful for controlling finite sets ${A}$ of small doubling, in the sense that ${|A+A| \leq K|A|}$ for some bounded ${K}$. (There are deeper theorems, most notably Freiman’s theorem, which give more control than what elementary Ruzsa calculus does, however the known bounds in the latter theorem are worse than polynomial in ${K}$ (although it is conjectured otherwise), whereas the elementary estimates are almost all polynomial in ${K}$.)

However, there are some settings in which the standard sum set estimates are not quite applicable. One such setting is the continuous setting, where one is dealing with bounded open sets in an additive Lie group (e.g. ${{\bf R}^n}$ or a torus ${({\bf R}/{\bf Z})^n}$) rather than a finite setting. Here, one can largely replicate the discrete sum set estimates by working with a Haar measure in place of cardinality; this is the approach taken for instance in this paper of mine. However, there is another setting, which one might dub the “discretised” setting (as opposed to the “discrete” setting or “continuous” setting), in which the sets ${A}$ remain finite (or at least discretisable to be finite), but for which there is a certain amount of “roundoff error” coming from the discretisation. As a typical example (working now in a non-commutative multiplicative setting rather than an additive one), consider the orthogonal group ${O_n({\bf R})}$ of orthogonal ${n \times n}$ matrices, and let ${A}$ be the matrices obtained by starting with all of the orthogonal matrice in ${O_n({\bf R})}$ and rounding each coefficient of each matrix in this set to the nearest multiple of ${\epsilon}$, for some small ${\epsilon>0}$. This forms a finite set (whose cardinality grows as ${\epsilon\rightarrow 0}$ like a certain negative power of ${\epsilon}$). In the limit ${\epsilon \rightarrow 0}$, the set ${A}$ is not a set of small doubling in the discrete sense. However, ${A \cdot A}$ is still close to ${A}$ in a metric sense, being contained in the ${O_n(\epsilon)}$-neighbourhood of ${A}$. Another key example comes from graphs ${\Gamma := \{ (x, f(x)): x \in G \}}$ of maps ${f: A \rightarrow H}$ from a subset ${A}$ of one additive group ${G = (G,+)}$ to another ${H = (H,+)}$. If ${f}$ is “approximately additive” in the sense that for all ${x,y \in G}$, ${f(x+y)}$ is close to ${f(x)+f(y)}$ in some metric, then ${\Gamma}$ might not have small doubling in the discrete sense (because ${f(x+y)-f(x)-f(y)}$ could take a large number of values), but could be considered a set of small doubling in a discretised sense.

One would like to have a sum set (or product set) theory that can handle these cases, particularly in “high-dimensional” settings in which the standard methods of passing back and forth between continuous, discrete, or discretised settings behave poorly from a quantitative point of view due to the exponentially large doubling constant of balls. One way to do this is to impose a translation invariant metric ${d}$ on the underlying group ${G = (G,+)}$ (reverting back to additive notation), and replace the notion of cardinality by that of metric entropy. There are a number of almost equivalent ways to define this concept:

Definition 3 Let ${(X,d)}$ be a metric space, let ${E}$ be a subset of ${X}$, and let ${r>0}$ be a radius.

• The packing number ${N^{pack}_r(E)}$ is the largest number of points ${x_1,\dots,x_n}$ one can pack inside ${E}$ such that the balls ${B(x_1,r),\dots,B(x_n,r)}$ are disjoint.
• The internal covering number ${N^{int}_r(E)}$ is the fewest number of points ${x_1,\dots,x_n \in E}$ such that the balls ${B(x_1,r),\dots,B(x_n,r)}$ cover ${E}$.
• The external covering number ${N^{ext}_r(E)}$ is the fewest number of points ${x_1,\dots,x_n \in X}$ such that the balls ${B(x_1,r),\dots,B(x_n,r)}$ cover ${E}$.
• The metric entropy ${N^{ent}_r(E)}$ is the largest number of points ${x_1,\dots,x_n}$ one can find in ${E}$ that are ${r}$-separated, thus ${d(x_i,x_j) \geq r}$ for all ${i \neq j}$.

It is an easy exercise to verify the inequalities

$\displaystyle N^{ent}_{2r}(E) \leq N^{pack}_r(E) \leq N^{ext}_r(E) \leq N^{int}_r(E) \leq N^{ent}_r(E)$

for any ${r>0}$, and that ${N^*_r(E)}$ is non-increasing in ${r}$ and non-decreasing in ${E}$ for the three choices ${* = pack,ext,ent}$ (but monotonicity in ${E}$ can fail for ${*=int}$!). It turns out that the external covering number ${N^{ent}_r(E)}$ is slightly more convenient than the other notions of metric entropy, so we will abbreviate ${N_r(E) = N^{ent}_r(E)}$. The cardinality ${|E|}$ can be viewed as the limit of the entropies ${N^*_r(E)}$ as ${r \rightarrow 0}$.

If we have the bounded doubling property that ${B(0,2r)}$ is covered by ${O(1)}$ translates of ${B(0,r)}$ for each ${r>0}$, and one has a Haar measure ${m}$ on ${G}$ which assigns a positive finite mass to each ball, then any of the above entropies ${N^*_r(E)}$ is comparable to ${m( E + B(0,r) ) / m(B(0,r))}$, as can be seen by simple volume packing arguments. Thus in the bounded doubling setting one can usually use the measure-theoretic sum set theory to derive entropy-theoretic sumset bounds (see e.g. this paper of mine for an example of this). However, it turns out that even in the absence of bounded doubling, one still has an entropy analogue of most of the elementary sum set theory, except that one has to accept some degradation in the radius parameter ${r}$ by some absolute constant. Such losses can be acceptable in applications in which the underlying sets ${A}$ are largely “transverse” to the balls ${B(0,r)}$, so that the ${N_r}$-entropy of ${A}$ is largely independent of ${A}$; this is a situation which arises in particular in the case of graphs ${\Gamma = \{ (x,f(x)): x \in G \}}$ discussed above, if one works with “vertical” metrics whose balls extend primarily in the vertical direction. (I hope to present a specific application of this type here in the near future.)

Henceforth we work in an additive group ${G}$ equipped with a translation-invariant metric ${d}$. (One can also generalise things slightly by allowing the metric to attain the values ${0}$ or ${+\infty}$, without changing much of the analysis below.) By the Heine-Borel theorem, any precompact set ${E}$ will have finite entropy ${N_r(E)}$ for any ${r>0}$. We now have analogues of the two basic Ruzsa lemmas above:

Lemma 4 (Ruzsa covering lemma) Let ${A, B}$ be precompact non-empty subsets of ${G}$, and let ${r>0}$. Then ${A}$ may be covered by at most ${\frac{N_r(A+B)}{N_r(B)}}$ translates of ${B-B+B(0,2r)}$.

Proof: Let ${a_1,\dots,a_n \in A}$ be a maximal set of points such that the sets ${a_i + B + B(0,r)}$ are all disjoint. Then the sets ${a_i+B}$ are disjoint in ${A+B}$ and have entropy ${N_r(a_i+B)=N_r(B)}$, and furthermore any ball of radius ${r}$ can intersect at most one of the ${a_i+B}$. We conclude that ${N_r(A+B) \geq n N_r(B)}$, so ${n \leq \frac{N_r(A+B)}{N_r(B)}}$. If ${a \in A}$, then ${a+B+B(0,r)}$ must intersect one of the ${a_i + B + B(0,r)}$, so ${a \in a_i + B-B + B(0,2r)}$, and the claim follows. $\Box$

Lemma 5 (Ruzsa triangle inequality) Let ${A,B,C}$ be precompact non-empty subsets of ${G}$, and let ${r>0}$. Then ${N_{4r}(A-C) \leq \frac{N_r(A-B) N_r(B-C)}{N_r(B)}}$.

Proof: Consider the addition map ${+: (x,y) \mapsto x+y}$ from ${(A-B) \times (B-C)}$ to ${G}$. The domain ${(A-B) \times (B-C)}$ may be covered by ${N_r(A-B) N_r(B-C)}$ product balls ${B(x,r) \times B(y,r)}$. Every element ${a-c}$ of ${A - C}$ has a preimage ${\{ (x,y) \in (A-B) \times (B-C)\}}$ of this map which projects to a translate of ${B}$, and thus must meet at least ${N_r(B)}$ of these product balls. However, if two elements of ${A-C}$ are separated by a distance of at least ${4r}$, then no product ball can intersect both preimages. We thus see that ${N_{4r}^{ent}(A-C) \leq \frac{N_r(A-B) N_r(B-C)}{N_r(A-C)}}$, and the claim follows. $\Box$

Below the fold we will record some further metric entropy analogues of sum set estimates (basically redoing much of Chapter 2 of my book with Van Vu). Unfortunately there does not seem to be a direct way to abstractly deduce metric entropy results from their sum set analogues (basically due to the failure of a certain strong version of Freiman’s theorem, as discussed in this previous post); nevertheless, the proofs of the discrete arguments are elementary enough that they can be modified with a small amount of effort to handle the entropy case. (In fact, there should be a very general model-theoretic framework in which both the discrete and entropy arguments can be processed in a unified manner; see this paper of Hrushovski for one such framework.)

It is also likely that many of the arguments here extend to the non-commutative setting, but for simplicity we will not pursue such generalisations here.