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I’ve just uploaded to the arXiv my article “Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory“, submitted to the new journal “EMS surveys in the mathematical sciences“. This is the first draft of a survey article on the polynomial method – a technique in combinatorics and number theory for controlling a relevant set of points by comparing it with the zero set of a suitably chosen polynomial, and then using tools from algebraic geometry (e.g. Bezout’s theorem) on that zero set. As such, the method combines algebraic geometry with combinatorial geometry, and could be viewed as the philosophy of a combined field which I dub “algebraic combinatorial geometry”. There is also an important extension of this method when one is working overthe reals, in which methods from algebraic topology (e.g. the ham sandwich theorem and its generalisation to polynomials), and not just algebraic geometry, come into play also.
The polynomial method has been used independently many times in mathematics; for instance, it plays a key role in the proof of Baker’s theorem in transcendence theory, or Stepanov’s method in giving an elementary proof of the Riemann hypothesis for finite fields over curves; in combinatorics, the nullstellenatz of Alon is also another relatively early use of the polynomial method. More recently, it underlies Dvir’s proof of the Kakeya conjecture over finite fields and Guth and Katz’s near-complete solution to the Erdos distance problem in the plane, and can be used to give a short proof of the Szemeredi-Trotter theorem. One of the aims of this survey is to try to present all of these disparate applications of the polynomial method in a somewhat unified context; my hope is that there will eventually be a systematic foundation for algebraic combinatorial geometry which naturally contains all of these different instances the polynomial method (and also suggests new instances to explore); but the field is unfortunately not at that stage of maturity yet.
This is something of a first draft, so comments and suggestions are even more welcome than usual. (For instance, I have already had my attention drawn to some additional uses of the polynomial method in the literature that I was not previously aware of.)
Van Vu and I have just uploaded to the arXiv our joint paper “Local universality of zeroes of random polynomials“. This paper is a sequel to our previous work on local universality of eigenvalues of (non-Hermitian) random matrices with independent entries. One can re-interpret these previous results as a universality result for a certain type of random polynomial , namely the characteristic polynomial of the random matrix . In this paper, we consider the analogous problem for a different model of random polynomial, namely polynomials with independent random coefficients. More precisely, we consider random polynomials of the form
where are deterministic complex coefficients, and are independent identically distributed copies of a complex random variable , which we normalise to have mean zero and variance one. For simplicity we will ignore the technical issue that the leading coefficient is allowed to vanish; then has zeroes (counting multiplicity), which can be viewed as a random point process in the complex plane. In analogy with other models (such as random matrix models), we expect the (suitably normalised) asymptotic statistics of this point process in the limit to be universal, in the sense that it is largely independent of the precise distribution of the atom variable .
Our results are fairly general with regard to the choice of coefficients , but we isolate three particular choice of coefficients that are particularly natural and well-studied in the literature:
- Flat polynomials (or Weyl polynomials) in which .
- Elliptic polynomials (or binomial polynomials) in which .
- Kac polynomials in which .
The flat and elliptic polynomials enjoy special symmetries in the model case when the atom distribution is a complex Gaussian . Indeed, the zeroes of elliptic polynomials with complex Gaussian coefficients have a distribution which is invariant with respect to isometries of the Riemann sphere (thus has the same distribution as ), while the zeroes of the limiting case of the flat polynomials with complex Gaussian coefficients are similarly invariant with respect to isometries of the complex plane . (For a nice geometric proof of this facts, I recommend the nice book of Hough, Krishnapur, Peres, and Virag.)
The global (i.e. coarse-scale) distribution of zeroes of these polynomials is well understood, first in the case of gaussian distributions using the fundamental tool of the Kac-Rice formula, and then for more general choices of atom distribution in the recent work of Kabluchko and Zaporozhets. The zeroes of the flat polynomials are asymptotically distributed according to the circular law, normalised to be uniformly distributed on the disk of radius centred at the origin. To put it a bit informally, the zeroes are asymptotically distributed according to the measure , where denotes Lebesgue measure on the complex plane. One can non-rigorously see the scale appear by observing that when is much larger than , we expect the leading term of the flat polynomial to dominate, so that the polynomial should not have any zeroes in this region.
Similarly, the distribution of the elliptic polynomials is known to be asymptotically distributed according to a Cauchy-type distribution . The Kac polynomials behave differently; the zeroes concentrate uniformly on the unit circle (which is reasonable, given that one would expect the top order term to dominate for and the bottom order term to dominate for ). In particular, whereas the typical spacing between zeroes in the flat and elliptic cases would be expected to be comparable to , the typical spacing between zeroes for a Kac polynomial would be expected instead to be comparable to .
In our paper we studied the local distribution of zeroes at the scale of the typical spacing. In the case of polynomials with continuous complex atom disribution , the natural statistic to measure here is the -point correlation function , which for distinct complex numbers can be defined as the probability that there is a zero in each of the balls for some infinitesimal , divided by the normalising factor . (One can also define a -point correlation function in the case of a discrete distribution, but it will be a measure rather than a function in that case.) Our first main theorem is a general replacement principle which asserts, very roughly speaking, that the asymptotic -point correlation functions of two random polynomials will agree if the log-magnitudes have asymptotically the same distribution (actually we have to consider the joint distribution of for several points , but let us ignore this detail for now), and if the polynomials obeys a “non-clustering property” which asserts, roughly speaking, that not too many of the zeroes of can typically concentrate in a small ball. This replacement principle was implicit in our previous paper (and can be viewed as a local-scale version of the global-scale replacement principle in this earlier paper of ours). Specialising the replacement principle to the elliptic or flat polynomials, and using the Lindeberg swapping argument, we obtain a Two Moment Theorem that asserts, roughly speaking, that the asymptotic behaviour of the -point correlation functions depends only on the first two moments of the real and imaginary components of , as long as one avoids some regions of space where universality is expected to break down. (In particular, because does not have a universal distribution, one does not expect universality to hold near the origin; there is a similar problem near infinity.) Closely related results, by a slightly different method, have also been obtained recently by Ledoan, Merkli, and Starr. A similar result holds for the Kac polynomials after rescaling to account for the narrower spacing between zeroes.
We also have analogous results in the case of polynomials with real coefficients (so that the coefficients and the atom distribution are both real). In this case one expects to see a certain number of real zeroes, together with conjugate pairs of complex zeroes. Instead of the -point correlation function , the natural object of study is now the mixed -point correlation function that (roughly speaking) controls how often one can find a real zero near the real numbers , and a complex zero near the points . It turns out that one can disentangle the real and strictly complex zeroes and obtain separate universality results for both zeroes, provided that at least one of the polynomials involved obeys a “weak repulsion estimate” that shows that the real zeroes do not cluster very close to each other (and that the complex zeroes do not cluster very close to their complex conjugates). Such an estimate is needed to avoid the situation of two nearby real zeroes “colliding” to create a (barely) complex zero and its complex conjugate, or the time reversal of such a collision. Fortunately, in the case of real gaussian polynomials one can use the Kac-Rice formula to establish such a weak repulsion estimate, allowing analogues of the above universality results for complex random polynomials in the real case. Among other things, this gives universality results for the number of real zeroes of a random flat or elliptic polynomial; it turns out this number is typically equal to and respectively. (For Kac polynomials, the situation is somewhat different; it was already known that thanks to a long series of papers, starting with the foundational work of Kac and culminating in the work of Ibragimov and Maslova.)
While our methods are based on our earlier work on eigenvalues of random matrices, the situation with random polynomials is actually somewhat easier to deal with. This is because the log-magnitude of a random polynomial with independent coefficients is significantly easier to control than the log-determinant of a random matrix, as the former can be controlled by the central limit theorem, while the latter requires significantly more difficult arguments (in particular, bounds on the least singular value combined with Girko’s Hermitization trick). As such, one could view the current paper as an introduction to our more complicated previous paper, and with this in mind we have written the current paper to be self-contained (though this did make the paper somewhat lengthy, checking in at 68 pages).
The purpose of this post is to link to a short unpublished note of mine that I wrote back in 2010 but forgot to put on my web page at the time. Entitled “A physical space proof of the bilinear Strichartz and local smoothing estimates for the Schrodinger equation“, it gives a proof of two standard estimates for the free (linear) Schrodinger equation in flat Euclidean space, namely the bilinear Strichartz estimate and the local smoothing estimate, using primarily “physical space” methods such as integration by parts, instead of “frequency space” methods based on the Fourier transform, although a small amount of Fourier analysis (basically sectoral projection to make the Schrodinger waves move roughly in a given direction) is still needed. This is somewhat in the spirit of an older paper of mine with Klainerman and Rodnianski doing something similar for the wave equation, and is also very similar to a paper of Planchon and Vega from 2009. The hope was that by avoiding the finer properties of the Fourier transform, one could obtain a more robust argument which could also extend to nonlinear, non-free, or non-flat situations. These notes were cited once or twice by some people that I had privately circulated them to, so I decided to put them online here for reference.
UPDATE, July 24: Fabrice Planchon has kindly supplied another note in which he gives a particularly simple proof of local smoothing in one dimension, and discusses some other variants of the method (related to the paper of Planchon and Vega cited earlier).
Tamar Ziegler and I have just uploaded to the arXiv our joint paper “A multi-dimensional Szemerédi theorem for the primes via a correspondence principle“. This paper is related to an earlier result of Ben Green and mine in which we established that the primes contain arbitrarily long arithmetic progressions. Actually, in that paper we proved a more general result:
Theorem 1 (Szemerédi’s theorem in the primes) Let be a subset of the primes of positive relative density, thus . Then contains arbitrarily long arithmetic progressions.
This result was based in part on an earlier paper of Green that handled the case of progressions of length three. With the primes replaced by the integers, this is of course the famous theorem of Szemerédi.
Szemerédi’s theorem has now been generalised in many different directions. One of these is the multidimensional Szemerédi theorem of Furstenberg and Katznelson, who used ergodic-theoretic techniques to show that any dense subset of necessarily contained infinitely many constellations of any prescribed shape. Our main result is to relativise that theorem to the primes as well:
Theorem 2 (Multidimensional Szemerédi theorem in the primes) Let , and let be a subset of the Cartesian power of the primes of positive relative density, thus
Then for any , contains infinitely many “constellations” of the form with and a positive integer.
In the case when is itself a Cartesian product of one-dimensional sets (in particular, if is all of ), this result already follows from Theorem 1, but there does not seem to be a similarly easy argument to deduce the general case of Theorem 2 from previous results. Simultaneously with this paper, an independent proof of Theorem 2 using a somewhat different method has been established by Cook, Maygar, and Titichetrakun.
The result is reminiscent of an earlier result of mine on finding constellations in the Gaussian primes (or dense subsets thereof). That paper followed closely the arguments of my original paper with Ben Green, namely it first enclosed (a W-tricked version of) the primes or Gaussian primes (in a sieve theoretic-sense) by a slightly larger set (or more precisely, a weight function ) of almost primes or almost Gaussian primes, which one could then verify (using methods closely related to the sieve-theoretic methods in the ongoing Polymath8 project) to obey certain pseudorandomness conditions, known as the linear forms condition and the correlation condition. Very roughly speaking, these conditions assert statements of the following form: if is a randomly selected integer, then the events of simultaneously being an almost prime (or almost Gaussian prime) are approximately independent for most choices of . Once these conditions are satisfied, one can then run a transference argument (initially based on ergodic-theory methods, but nowadays there are simpler transference results based on the Hahn-Banach theorem, due to Gowers and Reingold-Trevisan-Tulsiani-Vadhan) to obtain relative Szemerédi-type theorems from their absolute counterparts.
However, when one tries to adapt these arguments to sets such as , a new difficulty occurs: the natural analogue of the almost primes would be the Cartesian square of the almost primes – pairs whose entries are both almost primes. (Actually, for technical reasons, one does not work directly with a set of almost primes, but would instead work with a weight function such as that is concentrated on a set such as , but let me ignore this distinction for now.) However, this set does not enjoy as many pseudorandomness conditions as one would need for a direct application of the transference strategy to work. More specifically, given any fixed , and random , the four events
do not behave independently (as they would if were replaced for instance by the Gaussian almost primes), because any three of these events imply the fourth. This blocks the transference strategy for constellations which contain some right-angles to them (e.g. constellations of the form ) as such constellations soon turn into rectangles such as the one above after applying Cauchy-Schwarz a few times. (But a few years ago, Cook and Magyar showed that if one restricted attention to constellations which were in general position in the sense that any coordinate hyperplane contained at most one element in the constellation, then this obstruction does not occur and one can establish Theorem 2 in this case through the transference argument.) It’s worth noting that very recently, Conlon, Fox, and Zhao have succeeded in removing of the pseudorandomness conditions (namely the correlation condition) from the transference principle, leaving only the linear forms condition as the remaining pseudorandomness condition to be verified, but unfortunately this does not completely solve the above problem because the linear forms condition also fails for (or for weights concentrated on ) when applied to rectangular patterns.
There are now two ways known to get around this problem and establish Theorem 2 in full generality. The approach of Cook, Magyar, and Titichetrakun proceeds by starting with one of the known proofs of the multidimensional Szemerédi theorem – namely, the proof that proceeds through hypergraph regularity and hypergraph removal – and attach pseudorandom weights directly within the proof itself, rather than trying to add the weights to the result of that proof through a transference argument. (A key technical issue is that weights have to be added to all the levels of the hypergraph – not just the vertices and top-order edges – in order to circumvent the failure of naive pseudorandomness.) As one has to modify the entire proof of the multidimensional Szemerédi theorem, rather than use that theorem as a black box, the Cook-Magyar-Titichetrakun argument is lengthier than ours; on the other hand, it is more general and does not rely on some difficult theorems about primes that are used in our paper.
In our approach, we continue to use the multidimensional Szemerédi theorem (or more precisely, the equivalent theorem of Furstenberg and Katznelson concerning multiple recurrence for commuting shifts) as a black box. The difference is that instead of using a transference principle to connect the relative multidimensional Szemerédi theorem we need to the multiple recurrence theorem, we instead proceed by a version of the Furstenberg correspondence principle, similar to the one that connects the absolute multidimensional Szemerédi theorem to the multiple recurrence theorem. I had discovered this approach many years ago in an unpublished note, but had abandoned it because it required an infinite number of linear forms conditions (in contrast to the transference technique, which only needed a finite number of linear forms conditions and (until the recent work of Conlon-Fox-Zhao) a correlation condition). The reason for this infinite number of conditions is that the correspondence principle has to build a probability measure on an entire -algebra; for this, it is not enough to specify the measure of a single set such as , but one also has to specify the measure of “cylinder sets” such as where could be arbitrarily large. The larger gets, the more linear forms conditions one needs to keep the correspondence under control.
With the sieve weights we were using at the time, standard sieve theory methods could indeed provide a finite number of linear forms conditions, but not an infinite number, so my idea was abandoned. However, with my later work with Green and Ziegler on linear equations in primes (and related work on the Mobius-nilsequences conjecture and the inverse conjecture on the Gowers norm), Tamar and I realised that the primes themselves obey an infinite number of linear forms conditions, so one can basically use the primes (or a proxy for the primes, such as the von Mangoldt function ) as the enveloping sieve weight, rather than a classical sieve. Thus my old idea of using the Furstenberg correspondence principle to transfer Szemerédi-type theorems to the primes could actually be realised. In the one-dimensional case, this simply produces a much more complicated proof of Theorem 1 than the existing one; but it turns out that the argument works as well in higher dimensions and yields Theorem 2 relatively painlessly, except for the fact that it needs the results on linear equations in primes, the known proofs of which are extremely lengthy (and also require some of the transference machinery mentioned earlier). The problem of correlations in rectangles is avoided in the correspondence principle approach because one can compensate for such correlations by performing a suitable weighted limit to compute the measure of cylinder sets, with each requiring a different weighted correction. (This may be related to the Cook-Magyar-Titichetrakun strategy of weighting all of the facets of the hypergraph in order to recover pseudorandomness, although our contexts are rather different.)
Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our joint paper “Multiple recurrence and convergence results associated to -actions“. This paper is primarily concerned with limit formulae in the theory of multiple recurrence in ergodic theory. Perhaps the most basic formula of this type is the mean ergodic theorem, which (among other things) asserts that if is a measure-preserving -system (which, in this post, means that is a probability space and is measure-preserving and invertible, thus giving an action of the integers), and are functions, and is ergodic (which means that contains no -invariant functions other than the constants (up to almost everywhere equivalence, of course)), then the average
converges as to the expression
see e.g. this previous blog post. Informally, one can interpret this limit formula as an equidistribution result: if is drawn at random from (using the probability measure ), and is drawn at random from for some large , then the pair becomes uniformly distributed in the product space (using product measure ) in the limit as .
If we allow to be non-ergodic, then we still have a limit formula, but it is a bit more complicated. Let be the -invariant measurable sets in ; the -system can then be viewed as a factor of the original system , which is equivalent (in the sense of measure-preserving systems) to a trivial system (known as the invariant factor) in which the shift is trivial. There is then a projection map to the invariant factor which is a factor map, and the average (1) converges in the limit to the expression
where is the pushforward map associated to the map ; see e.g. this previous blog post. We can interpret this as an equidistribution result. If is a pair as before, then we no longer expect complete equidistribution in in the non-ergodic, because there are now non-trivial constraints relating with ; indeed, for any -invariant function , we have the constraint ; putting all these constraints together we see that (for almost every , at least). The limit (2) can be viewed as an assertion that this constraint are in some sense the “only” constraints between and , and that the pair is uniformly distributed relative to these constraints.
Limit formulae are known for multiple ergodic averages as well, although the statement becomes more complicated. For instance, consider the expression
for three functions ; this is analogous to the combinatorial task of counting length three progressions in various sets. For simplicity we assume the system to be ergodic. Naively one might expect this limit to then converge to
which would roughly speaking correspond to an assertion that the triplet is asymptotically equidistributed in . However, even in the ergodic case there can be additional constraints on this triplet that cannot be seen at the level of the individual pairs , . The key obstruction here is that of eigenfunctions of the shift , that is to say non-trivial functions that obey the eigenfunction equation almost everywhere for some constant (or -invariant) . Each such eigenfunction generates a constraint
tying together , , and . However, it turns out that these are in some sense the only constraints on that are relevant for the limit (3). More precisely, if one sets to be the sub-algebra of generated by the eigenfunctions of , then it turns out that the factor is isomorphic to a shift system known as the Kronecker factor, for some compact abelian group and some (irrational) shift ; the factor map pushes eigenfunctions forward to (affine) characters on . It is then known that the limit of (3) is
where is the closed subgroup
and is the Haar probability measure on ; see this previous blog post. The equation defining corresponds to the constraint (4) mentioned earlier. Among other things, this limit formula implies Roth’s theorem, which in the context of ergodic theory is the assertion that the limit (or at least the limit inferior) of (3) is positive when is non-negative and not identically vanishing.
If one considers a quadruple average
(analogous to counting length four progressions) then the situation becomes more complicated still, even in the ergodic case. In addition to the (linear) eigenfunctions that already showed up in the computation of the triple average (3), a new type of constraint also arises from quadratic eigenfunctions , which obey an eigenfunction equation in which is no longer constant, but is now a linear eigenfunction. For such functions, behaves quadratically in , and one can compute the existence of a constraint
between , , , and that is not detected at the triple average level. As it turns out, this is not the only type of constraint relevant for (5); there is a more general class of constraint involving two-step nilsystems which we will not detail here, but see e.g. this previous blog post for more discussion. Nevertheless there is still a similar limit formula to previous examples, involving a special factor which turns out to be an inverse limit of two-step nilsystems; this limit theorem can be extracted from the structural theory in this paper of Host and Kra combined with a limit formula for nilsystems obtained by Lesigne, but will not be reproduced here. The pattern continues to higher averages (and higher step nilsystems); this was first done explicitly by Ziegler, and can also in principle be extracted from the structural theory of Host-Kra combined with nilsystem equidistribution results of Leibman. These sorts of limit formulae can lead to various recurrence results refining Roth’s theorem in various ways; see this paper of Bergelson, Host, and Kra for some examples of this.
The above discussion was concerned with -systems, but one can adapt much of the theory to measure-preserving -systems for other discrete countable abelian groups , in which one now has a family of shifts indexed by rather than a single shift, obeying the compatibility relation . The role of the intervals in this more general setting is replaced by that of Folner sequences. For arbitrary countable abelian , the theory for double averages (1) and triple limits (3) is essentially identical to the -system case. But when one turns to quadruple and higher limits, the situation becomes more complicated (and, for arbitrary , still not fully understood). However one model case which is now well understood is the finite field case when is an infinite-dimensional vector space over a finite field (with the finite subspaces then being a good choice for the Folner sequence). Here, the analogue of the structural theory of Host and Kra was worked out by Vitaly, Tamar, and myself in these previous papers (treating the high characteristic and low characteristic cases respectively). In the finite field setting, it turns out that nilsystems no longer appear, and one only needs to deal with linear, quadratic, and higher order eigenfunctions (known collectively as phase polynomials). It is then natural to look for a limit formula that asserts, roughly speaking, that if is drawn at random from a -system and drawn randomly from a large subspace of , then the only constraints between are those that arise from phase polynomials. The main theorem of this paper is to establish this limit formula (which, again, is a little complicated to state explicitly and will not be done here). In particular, we establish for the first time that the limit actually exists (a result which, for -systems, was one of the main results of this paper of Host and Kra).
As a consequence, we can recover finite field analogues of most of the results of Bergelson-Host-Kra, though interestingly some of the counterexamples demonstrating sharpness of their results for -systems (based on Behrend set constructions) do not seem to be present in the finite field setting (cf. this previous blog post on the cap set problem). In particular, we are able to largely settle the question of when one has a Khintchine-type theorem that asserts that for any measurable set in an ergodic -system and any , one has
for a syndetic set of , where are distinct residue classes. It turns out that Khintchine-type theorems always hold for (and for ergodicity is not required), and for it holds whenever form a parallelogram, but not otherwise (though the counterexample here was such a painful computation that we ended up removing it from the paper, and may end up putting it online somewhere instead), and for larger we could show that the Khintchine property failed for generic choices of , though the problem of determining exactly the tuples for which the Khintchine property failed looked to be rather messy and we did not completely settle it.
Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our survey “Small doubling in groups“, for the proceedings of the upcoming Erdos Centennial. This is a short survey of the known results on classifying finite subsets of an (abelian) additive group or a (not necessarily abelian) multiplicative group that have small doubling in the sense that the sum set or product set is small. Such sets behave approximately like finite subgroups of (and there is a closely related notion of an approximate group in which the analogy is even tighter) , and so this subject can be viewed as a sort of approximate version of finite group theory. (Unfortunately, thus far the theory does not have much new to say about the classification of actual finite groups; progress has been largely made instead on classifying the (highly restricted) number of ways in which approximate groups can differ from a genuine group.)
In the classical case when is the integers , these sets were classified (in a qualitative sense, at least) by a celebrated theorem of Freiman, which roughly speaking says that such sets are necessarily “commensurate” in some sense with a (generalised) arithmetic progression of bounded rank. There are a number of essentially equivalent ways to define what “commensurate” means here; for instance, in the original formulation of the theorem, one asks that be a dense subset of , but in modern formulations it is often more convenient to require instead that be of comparable size to and be covered by a bounded number of translates of , or that and have an intersection that is of comparable size to both and (cf. the notion of commensurability in group theory).
Freiman’s original theorem was extended to more general abelian groups in a sequence of papers culminating in the paper of Green and Ruzsa that handled arbitrary abelian groups. As such groups now contain non-trivial finite subgroups, the conclusion of the theorem must be modified by allowing for “coset progressions” , which can be viewed as “extensions” of generalized arithmetic progressions by genuine finite groups .
The proof methods in these abelian results were Fourier-analytic in nature (except in the cases of sets of very small doubling, in which more combinatorial approaches can be applied, and there were also some geometric or combinatorial methods that gave some weaker structural results). As such, it was a challenge to extend these results to nonabelian groups, although for various important special types of ambient group (such as an linear group over a finite or infinite field) it turns out that one can use tools exploiting the special structure of those groups (e.g. for linear groups one would use tools from Lie theory and algebraic geometry) to obtain quite satisfactory results; see e.g. this survey of Pyber and Szabo for the linear case. When the ambient group is completely arbitrary, it turns out the problem is closely related to the classical Hilbert’s fifth problem of determining the minimal requirements of a topological group in order for such groups to have Lie structure; this connection was first observed and exploited by Hrushovski, and then used by Breuillard, Green, and myself to obtain the analogue of Freiman’s theorem for an arbitrary nonabelian group.
This survey is too short to discuss in much detail the proof techniques used in these results (although the abelian case is discussed in this book of mine with Vu, and the nonabelian case discussed in this more recent book of mine), but instead focuses on the statements of the various known results, as well as some remaining open questions in the subject (in particular, there is substantial work left to be done in making the estimates more quantitative, particularly in the nonabelian setting).
I’ve just uploaded to the arXiv my joint paper with Vitaly Bergelson, “Multiple recurrence in quasirandom groups“, which is submitted to Geom. Func. Anal.. This paper builds upon a paper of Gowers in which he introduced the concept of a quasirandom group, and established some mixing (or recurrence) properties of such groups. A -quasirandom group is a finite group with no non-trivial unitary representations of dimension at most . We will informally refer to a “quasirandom group” as a -quasirandom group with the quasirandomness parameter large (more formally, one can work with a sequence of -quasirandom groups with going to infinity). A typical example of a quasirandom group is where is a large prime. Quasirandom groups are discussed in depth in this blog post. One of the key properties of quasirandom groups established in Gowers’ paper is the following “weak mixing” property: if are subsets of , then for “almost all” , one has
where denotes the density of in . Here, we use to informally represent an estimate of the form (where is a quantity that goes to zero when the quasirandomness parameter goes to infinity), and “almost all ” denotes “for all in a subset of of density “. As a corollary, if have positive density in (by which we mean that is bounded away from zero, uniformly in the quasirandomness parameter , and similarly for ), then (if the quasirandomness parameter is sufficiently large) we can find elements such that , , . In fact we can find approximately such pairs . To put it another way: if we choose uniformly and independently at random from , then the events , , are approximately independent (thus the random variable resembles a uniformly distributed random variable on in some weak sense). One can also express this mixing property in integral form as
for any bounded functions . (Of course, with being finite, one could replace the integrals here by finite averages if desired.) Or in probabilistic language, we have
where are drawn uniformly and independently at random from .
As observed in Gowers’ paper, one can iterate this observation to find “parallelopipeds” of any given dimension in dense subsets of . For instance, applying (1) with replaced by , , and one can assert (after some relabeling) that for chosen uniformly and independently at random from , the events , , , , , , are approximately independent whenever are dense subsets of ; thus the tuple resebles a uniformly distributed random variable in in some weak sense.
However, there are other tuples for which the above iteration argument does not seem to apply. One of the simplest tuples in this vein is the tuple in , when are drawn uniformly at random from a quasirandom group . Here, one does not expect the tuple to behave as if it were uniformly distributed in , because there is an obvious constraint connecting the last two components of this tuple: they must lie in the same conjugacy class! In particular, if is a subset of that is the union of conjugacy classes, then the events , are perfectly correlated, so that is equal to rather than . Our main result, though, is that in a quasirandom group, this is (approximately) the only constraint on the tuple. More precisely, we have
Theorem 1 Let be a -quasirandom group, and let be drawn uniformly at random from . Then for any , we have
where goes to zero as , are drawn uniformly and independently at random from , and is drawn uniformly at random from the conjugates of for each fixed choice of .
This is the probabilistic formulation of the above theorem; one can also phrase the theorem in other formulations (such as an integral formulation), and this is detailed in the paper. This theorem leads to a number of recurrence results; for instance, as a corollary of this result, we have
for almost all , and any dense subsets of ; the lower and upper bounds are sharp, with the lower bound being attained when is randomly distributed, and the upper bound when is conjugation-invariant.
To me, the more interesting thing here is not the result itself, but how it is proven. Vitaly and I were not able to find a purely finitary way to establish this mixing theorem. Instead, we had to first use the machinery of ultraproducts (as discussed in this previous post) to convert the finitary statement about a quasirandom group to an infinitary statement about a type of infinite group which we call an ultra quasirandom group (basically, an ultraproduct of increasingly quasirandom finite groups). This is analogous to how the Furstenberg correspondence principle is used to convert a finitary combinatorial problem into an infinitary ergodic theory problem.
Ultra quasirandom groups come equipped with a finite, countably additive measure known as Loeb measure , which is very analogous to the Haar measure of a compact group, except that in the case of ultra quasirandom groups one does not quite have a topological structure that would give compactness. Instead, one has a slightly weaker structure known as a -topology, which is like a topology except that open sets are only closed under countable unions rather than arbitrary ones. There are some interesting measure-theoretic and topological issues regarding the distinction between topologies and -topologies (and between Haar measure and Loeb measure), but for this post it is perhaps best to gloss over these issues and pretend that ultra quasirandom groups come with a Haar measure. One can then recast Theorem 1 as a mixing theorem for the left and right actions of the ultra approximate group on itself, which roughly speaking is the assertion that
for “almost all” , if are bounded measurable functions on , with having zero mean on all conjugacy classes of , where are the left and right translation operators
To establish this mixing theorem, we use the machinery of idempotent ultrafilters, which is a particularly useful tool for understanding the ergodic theory of actions of countable groups that need not be amenable; in the non-amenable setting the classical ergodic averages do not make much sense, but ultrafilter-based averages are still available. To oversimplify substantially, the idempotent ultrafilter arguments let one establish mixing estimates of the form (2) for “many” elements of an infinite-dimensional parallelopiped known as an IP system (provided that the actions of this IP system obey some technical mixing hypotheses, but let’s ignore that for sake of this discussion). The claim then follows by using the quasirandomness hypothesis to show that if the estimate (2) failed for a large set of , then this large set would then contain an IP system, contradicting the previous claim.
Idempotent ultrafilters are an extremely infinitary type of mathematical object (one has to use Zorn’s lemma no fewer than three times just to construct one of these objects!). So it is quite remarkable that they can be used to establish a finitary theorem such as Theorem 1, though as is often the case with such infinitary arguments, one gets absolutely no quantitative control whatsoever on the error terms appearing in that theorem. (It is also mildly amusing to note that our arguments involve the use of ultrafilters in two completely different ways: firstly in order to set up the ultraproduct that converts the finitary mixing problem to an infinitary one, and secondly to solve the infinitary mixing problem. Despite some superficial similarities, there appear to be no substantial commonalities between these two usages of ultrafilters.) There is already a fair amount of literature on using idempotent ultrafilter methods in infinitary ergodic theory, and perhaps by further development of ultraproduct correspondence principles, one can use such methods to obtain further finitary consequences (although the state of the art for idempotent ultrafilter ergodic theory has not advanced much beyond the analysis of two commuting shifts currently, which is the main reason why our arguments only handle the pattern and not more sophisticated patterns).
We also have some miscellaneous other results in the paper. It turns out that by using the triangle removal lemma from graph theory, one can obtain a recurrence result that asserts that whenever is a dense subset of a finite group (not necessarily quasirandom), then there are pairs such that all lie in . Using a hypergraph generalisation of the triangle removal lemma known as the hypergraph removal lemma, one can obtain more complicated versions of this statement; for instance, if is a dense subset of , then one can find triples such that all lie in . But the method is tailored to the specific types of patterns given here, and we do not have a general method for obtaining recurrence or mixing properties for arbitrary patterns of words in some finite alphabet such as .
We also give some properties of a model example of an ultra quasirandom group, namely the ultraproduct of where is a sequence of primes going off to infinity. Thanks to the substantial recent progress (by Helfgott, Bourgain, Gamburd, Breuillard, and others) on understanding the expansion properties of the finite groups , we have a fair amount of knowledge on the ultraproduct as well; for instance any two elements of will almost surely generate a spectral gap. We don’t have any direct application of this particular ultra quasirandom group, but it might be interesting to study it further.
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