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Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our paper “A note on approximate subgroups of {GL_n({\bf C})} and uniformly nonamenable groups“. In this short note, we obtain a new proof of a “noncommutative Freiman” type theorem in linear groups {GL_n({\bf C})}. As discussed in earlier blog posts, a general question in additive (or multiplicative) combinatorics is to understand the structure of approximate groups – subsets {A} of genuine groups {G} which are a symmetric neighbourhood the identity (thus {id \in A} and {a^{-1} \in A} whenever {a \in A}), and such that the product set {A \cdot A := \{ ab: a,b \in A \}} is covered by {K} left (or right) translates of {A} for some bounded {K}. (The case {K=1} corresponds to the case of a genuine group.) Most of the focus in multiplicative combinatorics has been on the “discrete” case when {A} is a finite set, though continuous cases are also of interest (for instance, small balls around the identity in a Lie group are approximate groups).

In the discrete case, examples of approximate groups include:

  • Finite groups;
  • Balls in a discrete abelian group, or more generally a discrete nilpotent group, with boundedly many generators;
  • Extensions of the latter type of balls by finite groups;
  • Approximate groups {A} that are controlled by one of the previous examples {B}, in the sense that {A} has comparable cardinality to {B}, and can be covered by boundedly many translates of {B}.

It was conjectured independently by Helfgott and Lindenstrauss (private communication) that these are in fact the only examples of finite approximate groups. This conjecture is not yet settled in general (although we, with Tom Sanders, are making progress on this problem that we hope to be able to report on soon). However, many partial results are known. In particular, as part of the recent paper of Hrushovski in which model-theoretic techniques were introduced to study approximate groups, the following result was established:

Theorem 1 If {n=O(1)}, then every approximate subgroup of {GL_n({\bf C})} is controlled by a nilpotent approximate subgroup.

This result can be compared with Jordan’s theorem (discussed earlier on this blog) that every finite subgroup of {GL_n({\bf C})} is virtually abelian (with a uniform bound on the index of the abelian subgroup), or the special case of Gromov’s theorem for linear groups (which follows easily from the Tits alternative and the work of Milnor and of Wolf) that every finitely generated subgroup in {GL_n({\bf C})} of polynomial growth is virtually nilpotent.

Hrushovski’s proof of the above argument was quite sophisticated; one first transplants the problem using model-theoretic techniques to an infinitary setting, in which the approximate group induces a locally compact topological group structure, which can be played off against the Lie group structure of {GL_n({\bf C})} using the machinery of a paper of Larsen and Pink, as discussed in this previous blog article.

Two further proofs of this theorem were obtained by ourselves, as well as in the most recent version of a similar preprint by Pyber and Szabo. The arguments used here are variants of those used in earlier papers of Helfgott, and are based on establishing expansion of sets that generated Zariski-dense subgroups of various Lie groups (such as {SL_n({\bf C})}). Again, the machinery of Larsen and Pink (which controls how such approximate subgroups intersect with algebraic subgroups) plays a central role.

In this note we give a new proof of this theorem, based primarily on a different tool, namely the uniform Tits alternative of Breuillard. Recall that the Tits alternative asserts that a finitely generated subgroup of {GL_n({\bf C})} is either virtually solvable, or contains a copy of a free group on two generators. In other words, if {A} is a finite symmetric neighbourhood of the identity of {GL_n({\bf C})}, then either {A} generates a virtually solvable subgroup, or else some power {A^m} of {A} contains two elements {x,y} that generate a free group. As stated, {m} may depend on {A}. However, the uniform Tits alternative makes the stronger assertion that one can take {m=m(n)} to be uniform in {A}, and depend only on the dimension parameter {n}.

To use this alternative, we have the following simple observation, that asserts that multiplication by two elements that generate a free group forces a small amount of expansion:

Lemma 2 Let {A, B} be finite sets, such that {B} is symmetric and contains two elements {x,y} that generate a free group {F_2}. Then {|A \cdot B| \geq |A|}.

We remark that this lemma immediately establishes the classical fact that any group that contains a copy of {F_2} is not amenable, an observation initially made by von Neumann.

Proof: By foliating {A} into cosets of {F_2} and translating, we may assume without loss of generality that {A \subset F_2}. Observe that for every element {a} in {A}, at least three of the four elements {ax, ay, ax^{-1}, ay^{-1}} has a longer word length than {a}, while lying in {A \cdot X}. Furthermore, all such elements generated in this fashion are distinct (as one can recover the initial word {a} from the longer word by truncation). The claim follows. \Box

This can be combined with a lemma of Sanders (also independently established by Croot and Sisask), that asserts that for any approximate group {A}, and any {r=O(1)}, one can find a smaller version {S} of {A} – also a symmetric neighbourhood of the identity – with the property that {S^r \subset A^4}, while {S} remains of comparable size to {A}. (One should think of {A} as being like a ball of some radius {R}, in which case {S} is analogous to a ball of radius {R/r}). In particular, {A \cdot S^r \subset A^5} still has size comparable to {A}. Inspecting the size of the sets {A, A \cdot S, A \cdot S^2, \ldots, A \cdot S^r}, we conclude (if {r} is large enough) from the above lemma that {S} cannot contain two elements that generate a free group. Indeed, a slight modification of this argument shows that for any {m = O(1)}, if we take {r} sufficiently large depending on {m}, that {S^m} does not contain two elements that generate a free group. Applying the uniform Tits alternative, this shows that {S} generates a virtually solvable subgroup of {GL_n({\bf C})}. From the known product theory for such groups (due to Breuillard and Green), {S} (and hence {A}) is therefore controlled by a virtually nilpotent group, as desired.

Ben Green, Tamar Ziegler, and I have just uploaded to the arXiv our paper “An inverse theorem for the Gowers U^{s+1}[N] norm“, which was previously announced on this blog.  We are still planning one final round of reviewing the preprint before submitting the paper, but it has gotten to the stage where we are comfortable with having the paper available on the arXiv.

The main result of the paper is to establish the inverse conjecture for the Gowers norm over the integers, which has a number of applications, in particular to counting solutions to various linear equations in primes.  In spirit, the proof of the paper follows the 21-page announcement that was uploaded previously.  However, for various rather annoying technical reasons, the 117-page paper has to devote a large amount of space to setting up various bits of auxiliary machinery (as well as a dozen or so pages worth of examples and discussion).  For instance, the announcement motivates many of the steps of the argument by heuristically identifying nilsequences n \mapsto F(g(n) \Gamma) with bracket polynomial phases such as n \mapsto e( \{ \alpha n \} \beta n ).  However, a rather significant amount of theory (which was already worked out to a large extent by Leibman) is needed to formalise the “bracket algebra” needed to manipulate such bracket polynomials and to connect them with nilsequences.  Furthermore, the “piecewise smooth” nature of bracket polynomials causes some technical issues with the equidistribution theory for these sequences.  Our original version of the paper (which was even longer than the current version) set out this theory.  But we eventually decided that it was best to eschew almost all use of bracket polynomials (except as motivation and examples), and run the argument almost entirely within the language of nilsequences, to keep the argument a bit more notationally focused (and to make the equidistribution theory easier to establish).  But this was not without a tradeoff; some statements that are almost trivially true for bracket polynomials, required some “nilpotent algebra” to convert to the language of nilsequences.  Here are some examples of this:

  1. It is intuitively clear that a bracket polynomial phase e(P(n)) of degree k in one variable n can be “multilinearised” to a polynomial e(Q(n_1,\ldots,n_k)) of multi-degree (1,\ldots,1) in k variables n_1,\ldots,n_k, such that e(P(n)) and e(Q(n,\ldots,n)) agree modulo lower order terms.  For instance, if e(P(n)) = e(\alpha n \{ \beta n \{ \gamma n \} \}) (so k=3), then one could take e(Q(n_1,n_2,n_3)) = e( \alpha n_1 \{ \beta n_2 \{ \gamma n_3 \} \}).   The analogue of this statement for nilsequences is true, but required a moderately complicated nilpotent algebra construction using the Baker-Campbell-Hausdorff formula.
  2. Suppose one has a bracket polynomial phase e(P_h(n)) of degree k in one variable n that depends on an additional parameter h, in such a way that exactly one of the coefficients in each monomial depends on h.  Furthermore, suppose this dependence is bracket linear in h.  Then it is intuitively clear that this phase can be rewritten (modulo lower order terms) as e( Q(h,n) ) where Q is a bracket polynomial of multidegree (1,k) in (h,n).  For instance, if e(P_h(n)) = e( \{ \alpha_h n \} \beta n ) and \alpha_h = \{\gamma h \} \delta, then we can take e(Q(h,n)) = e(\{ \{\gamma h\} \delta n\} \beta n ).  The nilpotent algebra analogue of this claim is true, but requires another moderately complicated nilpotent algebra construction based on semi-direct products.
  3. A bracket polynomial has a fairly visible concept of a “degree” (analogous to the corresponding notion for true polynomials), as well as a “rank” (which, roughly speaking measures the number of parentheses in the bracket monomials, plus one).  Thus, for instance, the bracket monomial \{\{ \alpha n^4 \} \beta n \} \gamma n^2 has degree 7 and rank 3.  Defining degree and rank for nilsequences requires one to generalise the notion of a (filtered) nilmanifold to one in which the lower central series is replaced by a filtration indexed by both the degree and the rank.

There are various other tradeoffs of this type in this paper.  For instance, nonstandard analysis tools were introduced to eliminate what would otherwise be quite a large number of epsilons and regularity lemmas to manage, at the cost of some notational overhead; and the piecewise discontinuities mentioned earlier were eliminated by the use of vector-valued nilsequences, though this again caused some further notational overhead.    These difficulties may be a sign that we do not yet have the “right” proof of this conjecture, but one will probably have to wait a few years before we get a proper amount of perspective and understanding on this circle of ideas and results.

Ben Green, Tamar Ziegler, and I have just uploaded to the arXiv the note “An inverse theorem for the Gowers norm {U^{s+1}[N]} (announcement)“, not intended for publication. This is an announcement of our forthcoming solution of the inverse conjecture for the Gowers norm, which roughly speaking asserts that {U^{s+1}[N]} norm of a bounded function is large if and only if that function correlates with an {s}-step nilsequence of bounded complexity.

The full argument is quite lengthy (our most recent draft is about 90 pages long), but this is in large part due to the presence of various technical details which are necessary in order to make the argument fully rigorous. In this 20-page announcement, we instead sketch a heuristic proof of the conjecture, relying in a number of “cheats” to avoid the above-mentioned technical details. In particular:

  • In the announcement, we rely on somewhat vaguely defined terms such as “bounded complexity” or “linearly independent with respect to bounded linear combinations” or “equivalent modulo lower step errors” without specifying them rigorously. In the full paper we will use the machinery of nonstandard analysis to rigorously and precisely define these concepts.
  • In the announcement, we deal with the traditional linear nilsequences rather than the polynomial nilsequences that turn out to be better suited for finitary equidistribution theory, but require more notation and machinery in order to use.
  • In a similar vein, we restrict attention to scalar-valued nilsequences in the announcement, though due to topological obstructions arising from the twisted nature of the torus bundles used to build nilmanifolds, we will have to deal instead with vector-valued nilsequences in the main paper.
  • In the announcement, we pretend that nilsequences can be described by bracket polynomial phases, at least for the sake of making examples, although strictly speaking bracket polynomial phases only give examples of piecewise Lipschitz nilsequences rather than genuinely Lipschitz nilsequences.

With these cheats, it becomes possible to shorten the length of the argument substantially. Also, it becomes clearer that the main task is a cohomological one; in order to inductively deduce the inverse conjecture for a given step {s} from the conjecture for the preceding step {s-1}, the basic problem is to show that a certain (quasi-)cocycle is necessarily a (quasi-)coboundary. This in turn requires a detailed analysis of the top order and second-to-top order terms in the cocycle, which requires a certain amount of nilsequence equidistribution theory and additive combinatorics, as well as a “sunflower decomposition” to arrange the various nilsequences one encounters into a usable “normal form”.

It is often the case in modern mathematics that the informal heuristic way to explain an argument looks quite different (and is significantly shorter) than the way one would formally present the argument with all the details. This seems to be particularly true in this case; at a superficial level, the full paper has a very different set of notation than the announcement, and a lot of space is invested in setting up additional machinery that one can quickly gloss over in the announcement. We hope though that the announcement can provide a “road map” to help navigate the much longer paper to come.

Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our paper “Approximate subgroups of linear groups“, submitted to GAFA. This paper contains (the first part) of the results announced previously by us; the second part of these results, concerning expander groups, will appear subsequently. The release of this paper has been coordinated with the release of a parallel paper by Pyber and Szabo (previously announced within an hour(!) of our own announcement).

Our main result describes (with polynomial accuracy) the “sufficiently Zariski dense” approximate subgroups of simple algebraic groups {{\bf G}(k)}, or more precisely absolutely almost simple algebraic groups over {k}, such as {SL_d(k)}. More precisely, define a {K}-approximate subgroup of a genuine group {G} to be a finite symmetric neighbourhood of the identity {A} (thus {1 \in A} and {A^{-1}=A}) such that the product set {A \cdot A} can be covered by {K} left-translates (and equivalently, {K} right-translates) of {A}.

Let {k} be a field, and let {\overline{k}} be its algebraic closure. For us, an absolutely almost simple algebraic group over {k} is a linear algebraic group {{\bf G}(k)} defined over {k} (i.e. an algebraic subvariety of {GL_n(k)} for some {n} with group operations given by regular maps) which is connected (i.e. irreducible), and such that the completion {{\bf G}(\overline{k})} has no proper normal subgroups of positive dimension (i.e. the only normal subgroups are either finite, or are all of {{\bf G}(\overline{k})}. To avoid degeneracies we also require {{\bf G}} to be non-abelian (i.e. not one-dimensional). These groups can be classified in terms of their associated finite-dimensional simple complex Lie algebra, which of course is determined by its Dynkin diagram, together with a choice of weight lattice (and there are only finitely many such choices once the Lie algebra is fixed). However, the exact classification of these groups is not directly used in our work.

Our first main theorem classifies the approximate subgroups {A} of such a group {{\bf G}(k)} in the model case when {A} generates the entire group {{\bf G}(k)}, and {k} is finite; they are either very small or very large.

Theorem 1 (Approximate groups that generate) Let {{\bf G}(k)} be an absolutely almost simple algebraic group over {k}. If {k} is finite and {A} is a {K}-approximate subgroup of {{\bf G}(k)} that generates {{\bf G}(k)}, then either {|A| \leq K^{O(1)}} or {|A| \geq K^{-O(1)} |{\bf G}(k)|}, where the implied constants depend only on {{\bf G}}.

The hypothesis that {A} generates {{\bf G}(k)} cannot be removed completely, since if {A} was a proper subgroup of {{\bf G}(k)} of size intermediate between that of the trivial group and of {{\bf G}(k)}, then the conclusion would fail (with {K=O(1)}). However, one can relax the hypothesis of generation to that of being sufficiently Zariski-dense in {{\bf G}(k)}. More precisely, we have

Theorem 2 (Zariski-dense approximate groups) Let {{\bf G}(k)} be an absolutely almost simple algebraic group over {k}. If {A} is a {K}-approximate group) is not contained in any proper algebraic subgroup of {k} of complexity at most {M} (where {M} is sufficiently large depending on {{\bf G}}), then either {|A| \leq K^{O(1)}} or {|A| \geq K^{-O(1)} |\langle A \rangle|}, where the implied constants depend only on {{\bf G}} and {\langle A \rangle} is the group generated by {A}.

Here, we say that an algebraic variety has complexity at most {M} if it can be cut out of an ambient affine or projective space of dimension at most {M} by using at most {M} polynomials, each of degree at most {M}. (Note that this is not an intrinsic notion of complexity, but will depend on how one embeds the algebraic variety into an ambient space; but we are assuming that our algebraic group {{\bf G}(k)} is a linear group and thus comes with such an embedding.)

In the case when {k = {\bf C}}, the second option of this theorem cannot occur since {{\bf G}({\bf C})} is infinite, leading to a satisfactory classification of the Zariski-dense approximate subgroups of almost simple connected algebraic groups over {{\bf C}}. On the other hand, every approximate subgroup of {GL_n({\bf C})} is Zariski-dense in some algebraic subgroup, which can be then split as an extension of a semisimple algebraic quotient group by a solvable algebraic group (the radical of the Zariski closure). Pursuing this idea (and glossing over some annoying technical issues relating to connectedness), together with the Freiman theory for solvable groups over {{\bf C}} due to Breuillard and Green, we obtain our third theorem:

Theorem 3 (Freiman’s theorem in {GL_n({\bf C})}) Let {A} be a {K}-approximate subgroup of {GL_n({\bf C})}. Then there exists a nilpotent {K}-approximate subgroup {B} of size at most {K^{O(1)}|A|}, such that {A} is covered by {K^{O(1)}} translates of {B}.

This can be compared with Gromov’s celebrated theorem that any finitely generated group of polynomial growth is virtually nilpotent. Indeed, the above theorem easily implies Gromov’s theorem in the case of finitely generated subgroups of {GL_n({\bf C})}.

By fairly standard arguments, the above classification theorems for approximate groups can be used to give bounds on the expansion and diameter of Cayley graphs, for instance one can establish a conjecture of Babai and Seress that connected Cayley graphs on absolutely almost simple groups over a finite field have polylogarithmic diameter at most. Applications to expanders include the result on Suzuki groups mentioned in a previous post; further applications will appear in a forthcoming paper.

Apart from the general structural theory of algebraic groups, and some quantitative analogues of the basic theory of algebraic geometry (which we chose to obtain via ultrafilters, as discussed in this post), we rely on two basic tools. Firstly, we use a version of the pivot argument developed first by Konyagin and Bourgain-Glibichuk-Konyagin in the setting of sum-product estimates, and generalised to more non-commutative settings by Helfgott; this is discussed in this previous post. Secondly, we adapt an argument of Larsen and Pink (which we learned from a paper of Hrushovski) to obtain a sharp bound on the extent to which a sufficiently Zariski-dense approximate groups can concentrate in a (bounded complexity) subvariety; this is discussed at the end of this blog post.

Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv the paper “Suzuki groups as expanders“, to be submitted. The purpose of this paper is to finish off the last case of the following theorem:

Theorem 1 (All finite simple groups have expanders) For every finite simple non-abelian group {G}, there exists a set of generators {S} of cardinality bounded uniformly in {G}, such that the Cayley graph {\hbox{Cay}(G,S)} on {G} generated by {S} (i.e. the graph that connects {g} with {sg} for all {g \in G} and {s \in S}) has expansion constant bounded away from zero uniformly in {G}, or equivalently that {|A \cdot S| \geq (1+\epsilon) |A|} for all {A \subset G} with {|A| < |G|/2} and some {\epsilon>0} independent of {G}.

To put in an essentially equivalent way, one can quickly generate a random element of a finite simple group with a near-uniform distribution by multiplying together a few ({O(\log |G|)}, to be more precise) randomly chosen elements of a fixed set {S}. (The most well-known instance of this phenomenon is the famous result of Bayer and Diaconis that one can shuffle a 52-card deck reasonably well after seven riffle shuffles, and almost perfectly after ten.) Note that the abelian simple groups {{\bf Z}/p{\bf Z}} do not support expanders due to the slow mixing time of random walks in the abelian setting.

The first step in proving this theorem is, naturally enough, the classification of finite simple groups. The sporadic groups have bounded cardinality and are a trivial case of this theorem, so one only has to deal with the seventeen infinite families of finite non-abelian simple groups. With one exception, the groups {G} in all of these families contain a copy of {SL_2({\bf F}_q)} for some {q} that goes to infinity as {|G| \rightarrow \infty}. Using this and several other non-trivial tools (such as Kazhdan’s property (T) and a deep model-theoretic result of Hrushovski and Pillay), the above theorem was proven for all groups outside of this exceptional family by a series of works culminating in the paper of Kassabov, Lubotzky, and Nikolov.

The exceptional family is the family of Suzuki groups {Sz(q)}, where {q = 2^{2n+1}} is an odd power of {2}. The Suzuki group {Sz(q)} can be viewed explicitly as a subgroup of the symplectic group {Sp_4(q)} and has cardinality {q^2 (q^2+1)(q-1) \approx q^5}. This cardinality is not divisible by {3}, whereas all groups of the form {SL_2(k)} have cardinality divisible by {3}; thus Suzuki groups do not contain copies of {SL_2} and the Kassabov-Lubotsky-Nikolov argument does not apply.

Our main result is that the Suzuki groups also support expanders, thus completing the last case of the above theorem. In fact we can pick just two random elements {a, b} of the Suzuki group, and with probability {1-o_{q \rightarrow \infty}(1)}, the Cayley graph generated by {S = \{a,b,a^{-1},b^{-1}\}} will be an expander uniformly in {q}. (As stated in the paper of Kassabov-Lubotsky-Nikolov, the methods in that paper should give an upper bound on {S} which they conservatively estimate to be {1000}.)

Our methods are different, instead following closely the arguments of Bourgain and Gamburd, which established the analogue of our result (i.e. that two random elements generate an expander graph) for the family of groups {SL_2({\bf F}_p)} ({p} a large prime); the arguments there have since been generalised to several other major families of groups, and our result here can thus be viewed as one further such generalisation. Roughly speaking, the strategy is as follows. Let {\mu} be the uniform probability measure on the randomly chosen set of generators {S}, and let {\mu^{(n)}} be the {n}-fold convolution. We need {\mu^{(n)}} to converge rapidly to the uniform measure on {G} (with a mixing time of {O(\log |G|)}). There are three steps to obtain this mixing:

  • (Early period) When {n \sim c \log |G|} for some small {c > 0}, one wants {\mu^{(n)}} to spread out a little bit in the sense that no individual element of {G} is assigned a mass of any more than {|G|^{-\epsilon}} for some fixed {\epsilon > 0}. More generally, no proper subgroup {H} of {G} should be assigned a mass of more than {|G|^{-\epsilon}}.
  • (Middle period) Once {\mu^{(n)}} is somewhat spread out, one should be able to convolve this measure with itself a bounded number of times and conclude that the measure {\mu^{(Cn)}} for some suitable {C} is reasonably spread out in the sense that its {L^2} norm is comparable (up to powers of {|G|^{\epsilon}} for some small {\epsilon > 0}) to the {L^2} norm of the uniform distribution.
  • (Late period) Once {\mu^{(n)}} is reasonably spread out, a few more convolutions should make it extremely close to uniform (e.g. within {|G|^{-10}} in the {L^\infty} norm).

The late period claim is easy to establish from Gowers’ theory of quasirandom groups, the key point being that (like all other finite simple nonabelian groups), the Suzuki groups do not admit any non-trivial low-dimensional irreducible representations (we can for instance use a precise lower bound of {\gg q^{3/2}}, due to Landazuri and Seitz). (One can also proceed here using a trace formula argument of Sarnak and Xue; the two approaches are basically equivalent.) The middle period reduces, by a variant of the Balog-Szemerédi-Gowers lemma, to a product estimate in {Sz(q)} which was recently established by Pyber-Szábo and can also be obtained by the methods of proof of the results announced by ourselves. (These arguments are in turn based on an earlier result of Helfgott establishing the analogous claim for {SL_2({\bf F}_p)}.) This requires checking that {Sz(q)} is a “sufficiently Zariski dense” subgroup of the finite Lie group {Sp_4(q)}, but this can be done using an explicit description of the Suzuki group and the Schwartz-Zippel lemma.

The main difficulty is then to deal with the early period, obtaining some initial non-concentration in the random walk associated to {S} away from subgroups {H} of {Sz(q)}. These subgroups have been classified for some time (see e.g. the book of Wilson); they split into two families, the algebraic subgroups, which in the Suzuki case turn out to be solvable of derived length at most three, and the arithmetic subgroups, which are conjugate to {Sz(q_0)}, where {{\bf F}_{q_0}} is a subfield of {{\bf F}_q}.

In the algebraic case, one can prevent concentration using a lower bound on the girth of random Cayley graphs due to Gamburd, Hoory, Shahshahani, Shalev, and Virág (and we also provide an independent proof of this fact for completeness, which fortunately is able to avoid any really deep technology, such as Lang-Weil estimates); this follows an analogous argument of Bourgain-Gamburd in the {SL_2} case fairly closely, and is ultimately based on the fact that all the algebraic subgroups obey a fixed law (in this case, the law arises from the solvability). In the arithmetic case, the main task is to show that the coefficients of the characteristic polynomial of a typical word in {S} does not fall into a proper subfield of {{\bf F}_q}, but this can be accomplished by a variant of the Schwartz-Zippel lemma.

Ben Green, and I have just uploaded to the arXiv a short (six-page) paper “Yet another proof of Szemeredi’s theorem“, submitted to the 70th birthday conference proceedings for Endre Szemerédi. In this paper we put in print a folklore observation, namely that the inverse conjecture for the Gowers norm, together with the density increment argument, easily implies Szemerédi’s famous theorem on arithmetic progressions. This is unsurprising, given that Gowers’ proof of Szemerédi’s theorem proceeds through a weaker version of the inverse conjecture and a density increment argument, and also given that it is possible to derive Szemerédi’s theorem from knowledge of the characteristic factor for multiple recurrence (the ergodic theory analogue of the inverse conjecture, first established by Host and Kra), as was done by Bergelson, Leibman, and Lesigne (and also implicitly in the earlier paper of Bergelson, Host, and Kra); but to our knowledge the exact derivation of Szemerédi’s theorem from the inverse conjecture was not in the literature. Ordinarily this type of folklore might be considered too trifling (and too well known among experts in the field) to publish; but we felt that the venue of the Szemerédi birthday conference provided a natural venue for this particular observation.

The key point is that one can show (by an elementary argument relying primarily an induction on dimension argument and the Weyl recurrence theorem, i.e. that given any real {\alpha} and any integer {s \geq 1}, that the expression {\alpha n^s} gets arbitrarily close to an integer) that given a (polynomial) nilsequence {n \mapsto F(g(n)\Gamma)}, one can subdivide any long arithmetic progression (such as {[N] = \{1,\ldots,N\}}) into a number of medium-sized progressions, where the nilsequence is nearly constant on each progression. As a consequence of this and the inverse conjecture for the Gowers norm, if a set has no arithmetic progressions, then it must have an elevated density on a subprogression; iterating this observation as per the usual density-increment argument as introduced long ago by Roth, one obtains the claim. (This is very close to the scheme of Gowers’ proof.)

Technically, one might call this the shortest proof of Szemerédi’s theorem in the literature (and would be something like the sixteenth such genuinely distinct proof, by our count), but that would be cheating quite a bit, primarily due to the fact that it assumes the inverse conjecture for the Gowers norm, our current proof of which is checking in at about 100 pages…

Ben Green, and I have just uploaded to the arXiv our paper “An arithmetic regularity lemma, an associated counting lemma, and applications“, submitted (a little behind schedule) to the 70th birthday conference proceedings for Endre Szemerédi. In this paper we describe the general-degree version of the arithmetic regularity lemma, which can be viewed as the counterpart of the Szemerédi regularity lemma, in which the object being regularised is a function {f: [N] \rightarrow [0,1]} on a discrete interval {[N] = \{1,\ldots,N\}} rather than a graph, and the type of patterns one wishes to count are additive patterns (such as arithmetic progressions {n,n+d,\ldots,n+(k-1)d}) rather than subgraphs. Very roughly speaking, this regularity lemma asserts that all such functions can be decomposed as a degree {\leq s} nilsequence (or more precisely, a variant of a nilsequence that we call an virtual irrational nilsequence), plus a small error, plus a third error which is extremely tiny in the Gowers uniformity norm {U^{s+1}[N]}. In principle, at least, the latter two errors can be readily discarded in applications, so that the regularity lemma reduces many questions in additive combinatorics to questions concerning (virtual irrational) nilsequences. To work with these nilsequences, we also establish a arithmetic counting lemma that gives an integral formula for counting additive patterns weighted by such nilsequences.

The regularity lemma is a manifestation of the “dichotomy between structure and randomness”, as discussed for instance in my ICM article or FOCS article. In the degree {1} case {s=1}, this result is essentially due to Green. It is powered by the inverse conjecture for the Gowers norms, which we and Tamar Ziegler have recently established (paper to be forthcoming shortly; the {k=4} case of our argument is discussed here). The counting lemma is established through the quantitative equidistribution theory of nilmanifolds, which Ben and I set out in this paper.

The regularity and counting lemmas are designed to be used together, and in the paper we give three applications of this combination. Firstly, we give a new proof of Szemerédi’s theorem, which proceeds via an energy increment argument rather than a density increment one. Secondly, we establish a conjecture of Bergelson, Host, and Kra, namely that if {A \subset [N]} has density {\alpha}, and {\epsilon > 0}, then there exist {\gg_{\alpha,\epsilon} N} shifts {h} for which {A} contains at least {(\alpha^4 - \epsilon)N} arithmetic progressions of length {k=4} of spacing {h}. (The {k=3} case of this conjecture was established earlier by Green; the {k=5} case is false, as was shown by Ruzsa in an appendix to the Bergelson-Host-Kra paper.) Thirdly, we establish a variant of a recent result of Gowers-Wolf, showing that the true complexity of a system of linear forms over {[N]} indeed matches the conjectured value predicted in their first paper.

In all three applications, the scheme of proof can be described as follows:

  • Apply the arithmetic regularity lemma, and decompose a relevant function {f} into three pieces, {f_{nil}, f_{sml}, f_{unf}}.
  • The uniform part {f_{unf}} is so tiny in the Gowers uniformity norm that its contribution can be easily dealt with by an appropriate “generalised von Neumann theorem”.
  • The contribution of the (virtual, irrational) nilsequence {f_{nil}} can be controlled using the arithmetic counting lemma.
  • Finally, one needs to check that the contribution of the small error {f_{sml}} does not overwhelm the main term {f_{nil}}. This is the trickiest bit; one often needs to use the counting lemma again to show that one can find a set of arithmetic patterns for {f_{nil}} that is so sufficiently “equidistributed” that it is not impacted by the small error.

To illustrate the last point, let us give the following example. Suppose we have a set {A \subset [N]} of some positive density (say {|A| = 10^{-1} N}) and we have managed to prove that {A} contains a reasonable number of arithmetic progressions of length {5} (say), e.g. it contains at least {10^{-10} N^2} such progressions. Now we perturb {A} by deleting a small number, say {10^{-2} N}, elements from {A} to create a new set {A'}. Can we still conclude that the new set {A'} contains any arithmetic progressions of length {5}?

Unfortunately, the answer could be no; conceivably, all of the {10^{-10} N^2} arithmetic progressions in {A} could be wiped out by the {10^{-2} N} elements removed from {A}, since each such element of {A} could be associated with up to {|A|} (or even {5|A|}) arithmetic progressions in {A}.

But suppose we knew that the {10^{-10} N^2} arithmetic progressions in {A} were equidistributed, in the sense that each element in {A} belonged to the same number of such arithmetic progressions, namely {5 \times 10^{-9} N}. Then each element deleted from {A} only removes at most {5 \times 10^{-9} N} progressions, and so one can safely remove {10^{-2} N} elements from {A} and still retain some arithmetic progressions. The same argument works if the arithmetic progressions are only approximately equidistributed, in the sense that the number of progressions that a given element {a \in A} belongs to concentrates sharply around its mean (for instance, by having a small variance), provided that the equidistribution is sufficiently strong. Fortunately, the arithmetic regularity and counting lemmas are designed to give precisely such a strong equidistribution result.

A succinct (but slightly inaccurate) summation of the regularity+counting lemma strategy would be that in order to solve a problem in additive combinatorics, it “suffices to check it for nilsequences”. But this should come with a caveat, due to the issue of the small error above; in addition to checking it for nilsequences, the answer in the nilsequence case must be sufficiently “dispersed” in a suitable sense, so that it can survive the addition of a small (but not completely negligible) perturbation.

One last “production note”. Like our previous paper with Emmanuel Breuillard, we used Subversion to write this paper, which turned out to be a significant efficiency boost as we could work on different parts of the paper simultaneously (this was particularly important this time round as the paper was somewhat lengthy and complicated, and there was a submission deadline). When doing so, we found it convenient to split the paper into a dozen or so pieces (one for each section of the paper, basically) in order to avoid conflicts, and to help coordinate the writing process. I’m also looking into git (a more advanced version control system), and am planning to use it for another of my joint projects; I hope to be able to comment on the relative strengths of these systems (and with plain old email) in the future.

Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our announcement “Linear approximate groups“, submitted to Electronic Research Announcements.

The main result is a step towards the classification of {K}-approximate groups, in the specific setting of simple and semisimple Lie groups (with some partial results for more general Lie groups). For {K \geq 1}, define a {K}-approximate group to be a finite subset {A} of a group {G} which is a symmetric neighbourhood of the origin (thus {1 \in A} and {A^{-1} := \{a^{-1}: a \in A \}} is equal to {A}), and such that the product set {A \cdot A} is covered by {K} left-translates (or equivalently, {K} right-translates) of {A}. For {K=1}, this is the same concept as a finite subgroup of {G}, but for larger values of {K}, one also gets some interesting objects which are close to, but not exactly groups, such as geometric progressions {\{ g^n: -N \leq n \leq N \}} for some {g \in G} and {N \geq 1}.

The expectation is that {K}-approximate groups are {C_K}-controlled by “structured” objects, such as actual groups and progressions, though the precise formulation of this has not yet been finalised. (We say that one finite set {A} {K}-controls another {B} if {A} is at most {K} times larger than {B} in cardinality, and {B} can be covered by at most {K} left translates or right translates of {A}.) The task of stating and proving this statement is the noncommutative Freiman theorem problem, discussed in these earlier blog posts.

While this problem remains unsolved for general groups, significant progress has been made in special groups, notably abelian, nilpotent, and solvable groups. Furthermore, the work of Chang (over {{\mathbb C}}) and Helfgott (over {{\Bbb F}_p}) has established the important special cases of the special linear groups {SL_2(k)} and {SL_3(k)}:

Theorem 1 (Helfgott’s theorem) Let {d = 2,3} and let {k} be either {{\Bbb F}_p} or {{\mathbb C}} for some prime {p}. Let {A} be a {K}-approximate subgroup of {G = SL_d(k)}.

  • If {A} generates the entire group {SL_d(k)} (which is only possible in the finite case {k={\Bbb F}_p}), then {A} is either controlled by the trivial group or the whole group.
  • If {d=2}, then {A} is {K^{O(1)}}-controlled by a solvable {K^{O(1)}}-approximate subgroup {B} of {G}, or by {G} itself. If {k={\mathbb C}}, the latter possibility cannot occur, and {B} must be abelian.

Our main result is an extension of Helfgott’s theorem to {SL_d(k)} for general {d}. In fact, we obtain an analogous result for any simple (or almost simple) Chevalley group over an arbitrary finite field (not necessarily of prime order), or over {{\mathbb C}}. (Standard embedding arguments then allow us to in fact handle arbitrary fields.) The results from simple groups can also be extended to (almost) semisimple Lie groups by an approximate version of Goursat’s lemma. Given that general Lie groups are known to split as extensions of (almost) semisimple Lie groups by solvable Lie groups, and Freiman-type theorems are known for solvable groups also, this in principle gives a Freiman-type theorem for arbitrary Lie groups; we have already established this in the characteristic zero case {k={\mathbb C}}, but there are some technical issues in the finite characteristic case {k = {\Bbb F}_q} that we are currently in the process of resolving.

We remark that a qualitative version of this result (with the polynomial bounds {K^{O(1)}} replaced by an ineffective bound {O_K(1)}) was also recently obtained by Hrushovski.

Our arguments are based in part on Helfgott’s arguments, in particular maximal tori play a major role in our arguments for much the same reason they do in Helfgott’s arguments. Our main new ingredient is a surprisingly simple argument, which we call the pivot argument, which is an analogue of a corresponding argument of Konyagin and Bourgain-Glibichuk-Konyagin that was used to prove a sum-product estimate. Indeed, it seems that Helfgott-type results in these groups can be viewed as a manifestation of a product-conjugation phenomenon analogous to the sum-product phenomenon. Namely, the sum-product phenomenon asserts that it is difficult for a subset of a field to be simultaneously approximately closed under sums and products, without being close to an actual field; similarly, the product-conjugation phenomenon asserts that it is difficult for a union of (subsets of) tori to be simultaneously approximately closed under products and conjugations, unless it is coming from a genuine group. In both cases, the key is to exploit a sizeable gap between the behaviour of two types of “pivots” (which are scaling parameters {\xi} in the sum-product case, and tori in the product-conjugation case): ones which interact strongly with the underlying set {A}, and ones which do not interact at all. The point is that there is no middle ground of pivots which only interact weakly with the set. This separation between interacting (or “involved”) and non-interacting (or “non-involved”) pivots can then be exploited to bootstrap approximate algebraic structure into exact algebraic structure. (Curiously, a similar argument is used all the time in PDE, where it goes under the name of the “bootstrap argument”.)

Below the fold we give more details of this crucial pivot argument.

One piece of trivia about the writing of this paper: this was the first time any of us had used modern version control software to collaboratively write a paper; specifically, we used Subversion, with the repository being hosted online by xp-dev. (See this post at the Secret Blogging Seminar for how to get started with this software.) There were a certain number of technical glitches in getting everything to install and run smoothly, but once it was set up, it was significantly easier to use than our traditional system of emailing draft versions of the paper back and forth, as one could simply download and upload the most recent versions whenever one wished, with all changes merged successfully. I had a positive impression of this software and am likely to try it again in future collaborations, particularly those involving at least three people. (It would also work well for polymath projects, modulo the technical barrier of every participant having to install some software.)

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This is an adaptation of a talk I gave recently for a program at IPAM. In this talk, I gave a (very informal and non-rigorous) overview of Hrushovski’s use of model-theoretic techniques to establish new Freiman-type theorems in non-commutative groups, and some recent work in progress of Ben Green, Tom Sanders and myself to establish combinatorial proofs of some of Hrushovski’s results.

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Ben Green, Tamar Ziegler and I have just uploaded to the arXiv our paper “An inverse theorem for the Gowers U^4 norm“.  This paper establishes the next case of the inverse conjecture for the Gowers norm for the integers (after the U^3 case, which was done by Ben and myself a few years ago).  This conjecture has a number of combinatorial and number-theoretic consequences, for instance by combining this new inverse theorem with previous results, one can now get the correct asymptotic for the number of arithmetic progressions of primes of length five in any large interval [N] = \{1,\ldots,N\}.

To state the inverse conjecture properly requires a certain amount of notation.  Given a function f: {\Bbb Z} \to {\Bbb C} and a shift h \in {\Bbb Z}, define the multiplicative derivative

\Delta_h f(x) := f(x+h) \overline{f(x)}

and then define the Gowers U^{s+1}[N] norm of a function f: [N] \to {\Bbb C} to (essentially) be the quantity

\| f\|_{U^{s+1}[N]} := ({\Bbb E}_{h_1,\ldots,h_{s+1} \in [-N,N]} {\Bbb E}_{x \in [N]} |\Delta_{h_1} \ldots \Delta_{h_{s+1}} f(x)|)^{1/2^{s+1}},

where we extend f by zero outside of [N].  (Actually, we use a slightly different normalisation to ensure that the function 1 has a U^{s+1} norm of 1, but never mind this for now.)

Informally, the Gowers norm \|f\|_{U^{s+1}[N]} measures the amount of bias present in the (s+1)^{st} multiplicative derivatives of f.  In particular, if f = e(P) := e^{2\pi i P} for some polynomial P: {\Bbb Z} \to {\Bbb C}, then the (s+1)^{th} derivative of f is identically 1, and so is the Gowers norm.

However, polynomial phases are not the only functions with large Gowers norm.  For instance, consider the function f(n) := e( \lfloor \sqrt{2} n \rfloor \sqrt{3} n ), which is what we call a quadratic bracket polynomial phase.  This function isn’t quite quadratic, but it is close enough to being quadratic (because one has the approximate linearity relationship \lfloor x+y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor holding a good fraction of the time) that it turns out that third derivative is trivial fairly often, and the Gowers norm \|f\|_{U^3[N]} is comparable to 1.  This bracket polynomial phase can be modeled as a nilsequence n \mapsto F( g(n) \Gamma ), where n \mapsto g(n) \Gamma is a polynomial orbit on a nilmanifold G/\Gamma, which in this case has step 2.  (The function F is only piecewise smooth, due to the discontinuity in the floor function \lfloor \rfloor, so strictly speaking we would classify this as an almost nilsequence rather than a nilsequence, but let us ignore this technical issue here.)  In fact, there is a very close relationship between nilsequences and bracket polynomial phases, but I will detail this in a later post.

The inverse conjecture for the Gowers norm, GI(s), asserts that such nilsequences are the only obstruction to the Gowers norm being small.  Roughly speaking, it goes like this:

Inverse conjecture, GI(s). (Informal statement)  Suppose that f: [N] \to {\Bbb C} is bounded but has large U^{s+1}[N] norm.  Then there is an s-step nilsequence n \mapsto F( g(n) \Gamma ) of “bounded complexity” that correlates with f.

This conjecture is trivial for s=0, is a short consequence of Fourier analysis when s=1, and was proven for s=2 by Ben and myself.  In this paper we establish the s=3 case.  An equivalent formulation in this case is that any bounded function f of large U^4 norm must correlate with a “bracket cubic phase”, which is the product of a bounded number of phases from the following list

e( \alpha n^3 + \beta n^2 + \gamma n), e( \lfloor \alpha n \rfloor \beta n^2 ), e( \lfloor \alpha n \rfloor \lfloor \beta n \rfloor \gamma n ), e( \lfloor \alpha n \rfloor \beta n ) (*)

for various real numbers \alpha,\beta,\gamma.

It appears that our methods also work in higher step, though for technical reasons it is convenient to make a number of adjustments to our arguments to do so, most notably a switch from standard analysis to non-standard analysis, about which I hope to say more later.  But there are a number of simplifications available on the s=3 case which make the argument significantly shorter, and so we will be writing the higher s argument in a separate paper.

The arguments largely follow those for the s=2 case (which in turn are based on this paper of Gowers).  Two major new ingredients are a deployment of a normal form and equidistribution theory for bracket quadratic phases, and a combinatorial decomposition of frequency space which we call the sunflower decomposition.  I will sketch these ideas below the fold.

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