You are currently browsing the category archive for the ‘paper’ category.

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 1Let be a -quasirandom group, and let be drawn uniformly at random from . Then for any , we havewhere 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.

I recently finished the first draft of the last of my books based on my 2011 blog posts (and also my Google buzzes and Google+ posts from that year), entitled “Spending symmetry“. The PDF of this draft is available here. This is again a rather assorted (and lightly edited) collection of posts (and buzzes, and Google+ posts), though concentrating in the areas of analysis (both standard and nonstandard), logic, and geometry. As always, comments and corrections are welcome.

Ben Green and I have just uploaded to the arXiv our new paper “On sets defining few ordinary lines“, submitted to Discrete and Computational Geometry. This paper asymptotically solves two old questions concerning finite configurations of points in the plane . Given a set of points in the plane, define an *ordinary line* to be a line containing exactly two points of . The classical Sylvester-Gallai theorem, first posed as a problem by Sylvester in 1893, asserts that as long as the points of are not all collinear, defines at least one ordinary line:

It is then natural to pose the question of what is the minimal number of ordinary lines that a set of non-collinear points can generate. In 1940, Melchior gave an elegant proof of the Sylvester-Gallai theorem based on projective duality and Euler’s formula , showing that at least three ordinary lines must be created; in 1951, Motzkin showed that there must be ordinary lines. Previously to this paper, the best lower bound was by Csima and Sawyer, who in 1993 showed that there are at least ordinary lines. In the converse direction, if is even, then by considering equally spaced points on a circle, and points on the line at infinity in equally spaced directions, one can find a configuration of points that define just ordinary lines.

As first observed by Böröczky, variants of this example also give few ordinary lines for odd , though not quite as few as ; more precisely, when one can find a configuration with ordinary lines, and when one can find a configuration with ordinary lines. Our first main result is that these configurations are best possible for sufficiently large :

Theorem 1 (Dirac-Motzkin conjecture)If is sufficiently large, then any set of non-collinear points in the plane will define at least ordinary lines. Furthermore, if is odd, at least ordinary lines must be created.

The *Dirac-Motzkin conjecture* asserts that the first part of this theorem in fact holds for all , not just for sufficiently large ; in principle, our theorem reduces that conjecture to a finite verification, although our bound for “sufficiently large” is far too poor to actually make this feasible (it is of double exponential type). (There are two known configurations for which one has ordinary lines, one with (discovered by Kelly and Moser), and one with (discovered by Crowe and McKee).)

Our second main result concerns not the ordinary lines, but rather the *-rich lines* of an -point set – a line that meets exactly three points of that set. A simple double counting argument (counting pairs of distinct points in the set in two different ways) shows that there are at most

-rich lines. On the other hand, on an elliptic curve, three distinct points P,Q,R on that curve are collinear precisely when they sum to zero with respect to the group law on that curve. Thus (as observed first by Sylvester in 1868), any finite subgroup of an elliptic curve (of which one can produce numerous examples, as elliptic curves in have the group structure of either or ) can provide examples of -point sets with a large number of -rich lines (, to be precise). One can also shift such a finite subgroup by a third root of unity and obtain a similar example with only one fewer -rich line. Sylvester then formally posed the question of determining whether this was best possible.

This problem was known as the Orchard planting problem, and was given a more poetic formulation as such by Jackson in 1821 (nearly fifty years prior to Sylvester!):

Our second main result answers this problem affirmatively in the large case:

Theorem 2 (Orchard planting problem)If is sufficiently large, then any set of points in the plane will determine at most -rich lines.

Again, our threshold for “sufficiently large” for this is extremely large (though slightly less large than in the previous theorem), and so a full solution of the problem, while in principle reduced to a finitary computation, remains infeasible at present.

Our results also classify the extremisers (and near extremisers) for both of these problems; basically, the known examples mentioned previously are (up to projective transformation) the only extremisers when is sufficiently large.

Our proof strategy follows the “inverse theorem method” from additive combinatorics. Namely, rather than try to prove *direct* theorems such as lower bounds on the number of ordinary lines, or upper bounds on the number of -rich lines, we instead try to prove *inverse* theorems (also known as *structure theorems*), in which one attempts a structural classification of all configurations with very few ordinary lines (or very many -rich lines). In principle, once one has a sufficiently explicit structural description of these sets, one simply has to compute the precise number of ordinary lines or -rich lines in each configuration in the list provided by that structural description in order to obtain results such as the two theorems above.

Note from double counting that sets with many -rich lines will necessarily have few ordinary lines. Indeed, if we let denote the set of lines that meet exactly points of an -point configuration, so that is the number of -rich lines and is the number of ordinary lines, then we have the double counting identity

which among other things implies that any counterexample to the orchard problem can have at most ordinary lines. In particular, any structural theorem that lets us understand configurations with ordinary lines will, in principle, allow us to obtain results such as the above two theorems.

As it turns out, we do eventually obtain a structure theorem that is strong enough to achieve these aims, but it is difficult to prove this theorem directly. Instead we proceed more iteratively, beginning with a “cheap” structure theorem that is relatively easy to prove but provides only a partial amount of control on the configurations with ordinary lines. One then builds upon that theorem with additional arguments to obtain an “intermediate” structure theorem that gives better control, then a “weak” structure theorem that gives even more control, a “strong” structure theorem that gives yet more control, and then finally a “very strong” structure theorem that gives an almost complete description of the configurations (but only in the asymptotic regime when is very large). It turns out that the “weak” theorem is enough for the orchard planting problem, and the “strong” version is enough for the Dirac-Motzkin conjecture. (So the “very strong” structure theorem ends up being unnecessary for the two applications given, but may be of interest for other applications.) Note that the stronger theorems do not completely supercede the weaker ones, because the quantitative bounds in the theorems get progressively worse as the control gets stronger.

Before we state these structure theorems, note that all the examples mentioned previously of sets with few ordinary lines involved cubic curves: either irreducible examples such as elliptic curves, or reducible examples such as the union of a circle (or more generally, a conic section) and a line. (We also allow singular cubic curves, such as the union of a conic section and a tangent line, or a singular irreducible curve such as .) This turns out to be no coincidence; cubic curves happen to be very good at providing many -rich lines (and thus, few ordinary lines), and conversely it turns out that they are essentially the only way to produce such lines. This can already be evidenced by our cheap structure theorem:

Theorem 3 (Cheap structure theorem)Let be a configuration of points with at most ordinary lines for some . Then can be covered by at most cubic curves.

This theorem is already a non-trivial amount of control on sets with few ordinary lines, but because the result does not specify the nature of these curves, and how they interact with each other, it does not seem to be directly useful for applications. The intermediate structure theorem given below gives a more precise amount of control on these curves (essentially guaranteeing that all but at most one of the curve components are lines):

Theorem 4 (Intermediate structure theorem)Let be a configuration of points with at most ordinary lines for some . Then one of the following is true:

- lies on the union of an irreducible cubic curve and an additional points.
- lies on the union of an irreducible conic section and an additional lines, with of the points on in either of the two components.
- lies on the union of lines and an additional points.

By some additional arguments (including a very nice argument supplied to us by Luke Alexander Betts, an undergraduate at Cambridge, which replaces a much more complicated (and weaker) argument we originally had for this paper), one can cut down the number of lines in the above theorem to just one, giving a more useful structure theorem, at least when is large:

Theorem 5 (Weak structure theorem)Let be a configuration of points with at most ordinary lines for some . Assume that for some sufficiently large absolute constant . Then one of the following is true:

- lies on the union of an irreducible cubic curve and an additional points.
- lies on the union of an irreducible conic section, a line, and an additional points, with of the points on in either of the first two components.
- lies on the union of a single line and an additional points.

As mentioned earlier, this theorem is already strong enough to resolve the orchard planting problem for large . The presence of the double exponential here is extremely annoying, and is the main reason why the final thresholds for “sufficiently large” in our results are excessively large, but our methods seem to be unable to eliminate these exponentials from our bounds (though they can fortunately be confined to a lower bound for , keeping the other bounds in the theorem polynomial in ).

For the Dirac-Motzkin conjecture one needs more precise control on the portion of on the various low-degree curves indicated. This is given by the following result:

Theorem 6 (Strong structure theorem)Let be a configuration of points with at most ordinary lines for some . Assume that for some sufficiently large absolute constant . Then, after adding or deleting points from if necessary (modifying appropriately), and then applying a projective transformation, one of the following is true:

- is a finite subgroup of an elliptic curve (EDIT: as pointed out in comments, one also needs to allow for finite subgroups of acnodal singular cubic curves), possibly shifted by a third root of unity.
- is the Borozcky example mentioned previously (the union of equally spaced points on the circle, and points on the line at infinity).
- lies on a single line.

By applying a final “cleanup” we can replace the in the above theorem with the optimal , which is our “very strong” structure theorem. But the strong structure theorem is already sufficient to establish the Dirac-Motzkin conjecture for large .

There are many tools that go into proving these theorems, some of which are extremely classical (with at least one going back to the ancient Greeks), and others being more recent. I will discuss some (not all) of these tools below the fold, and more specifically:

- Melchior’s argument, based on projective duality and Euler’s formula, initially used to prove the Sylvester-Gallai theorem;
- Chasles’ version of the Cayley-Bacharach theorem, which can convert dual triangular grids (produced by Melchior’s argument) into cubic curves that meet many points of the original configuration );
- Menelaus’s theorem, which is useful for producing ordinary lines when the point configuration lies on a few
*non-concurrent*lines, particularly when combined with a sum-product estimate of Elekes, Nathanson, and Ruzsa; - Betts’ argument, that produces ordinary lines when the point configuration lies on a few
*concurrent*lines; - A result of Poonen and Rubinstein that any point not on the origin or unit circle can lie on at most seven chords connecting roots of unity; this, together with a variant for elliptic curves, gives the very strong structure theorem, and is also (a strong version of) what is needed to finish off the Dirac-Motzkin and orchard planting problems from the structure theorems given above.

There are also a number of more standard tools from arithmetic combinatorics (e.g. a version of the Balog-Szemeredi-Gowers lemma) which are needed to tie things together at various junctures, but I won’t say more about these methods here as they are (by now) relatively routine.

Two quick updates with regards to polymath projects. Firstly, given the poll on starting the mini-polymath4 project, I will start the project at Thu July 12 2012 UTC 22:00. As usual, the main research thread on this project will be held at the polymath blog, with the discussion thread hosted separately on this blog.

Second, the Polymath7 project, which seeks to establish the “hot spots conjecture” for acute-angled triangles, has made a fair amount of progress so far; for instance, the first part of the conjecture (asserting that the second Neumann eigenfunction of an acute non-equilateral triangle is simple) is now solved, and the second part (asserting that the “hot spots” (i.e. extrema) of that second eigenfunction lie on the boundary of the triangle) has been solved in a number of special cases (such as the isosceles case). It’s been quite an active discussion in the last week or so, with almost 200 comments across two threads (and a third thread freshly opened up just now). While the problem is still not completely solved, I feel optimistic that it should fall within the next few weeks (if nothing else, it seems that the problem is now at least amenable to a brute force numerical attack, though personally I would prefer to see a more conceptual solution).

Van Vu and I have just uploaded to the arXiv our paper “Random matrices: Universality of local spectral statistics of non-Hermitian matrices“. The main result of this paper is a “Four Moment Theorem” that establishes universality for local spectral statistics of *non-Hermitian* matrices with independent entries, under the additional hypotheses that the entries of the matrix decay exponentially, and match moments with either the real or complex gaussian ensemble to fourth order. This is the non-Hermitian analogue of a long string of recent results establishing universality of local statistics in the Hermitian case (as discussed for instance in this recent survey of Van and myself, and also in several other places).

The complex case is somewhat easier to describe. Given a (non-Hermitian) random matrix ensemble of matrices, one can arbitrarily enumerate the (geometric) eigenvalues as , and one can then define the -point correlation functions to be the symmetric functions such that

In the case when is drawn from the complex gaussian ensemble, so that all the entries are independent complex gaussians of mean zero and variance one, it is a classical result of Ginibre that the asymptotics of near some point as and is fixed are given by the determinantal rule

for , where is the reproducing kernel

(There is also an asymptotic for the boundary case , but it is more complicated to state.) In particular, we see that for almost every , which is a manifestation of the well-known *circular law* for these matrices; but the circular law only captures the macroscopic structure of the spectrum, whereas the asymptotic (1) describes the microscopic structure.

Our first main result is that the asymptotic (1) for also holds (in the sense of vague convergence) when is a matrix whose entries are independent with mean zero, variance one, exponentially decaying tails, and which all match moments with the complex gaussian to fourth order. (Actually we prove a stronger result than this which is valid for all bounded and has more uniform bounds, but is a bit more technical to state.) An analogous result is also established for real gaussians (but now one has to separate the correlation function into components depending on how many eigenvalues are real and how many are strictly complex; also, the limiting distribution is more complicated, being described by Pfaffians rather than determinants). Among other things, this allows us to partially extend some known results on complex or real gaussian ensembles to more general ensembles. For instance, there is a central limit theorem of Rider which establishes a central limit theorem for the number of eigenvalues of a complex gaussian matrix in a mesoscopic disk; from our results, we can extend this central limit theorem to matrices that match the complex gaussian ensemble to fourth order, provided that the disk is small enough (for technical reasons, our error bounds are not strong enough to handle large disks). Similarly, extending some results of Edelman-Kostlan-Shub and of Forrester-Nagao, we can show that for a matrix matching the real gaussian ensemble to fourth order, the number of real eigenvalues is with probability for some absolute constant .

There are several steps involved in the proof. The first step is to apply the *Girko Hermitisation trick* to replace the problem of understanding the spectrum of a non-Hermitian matrix, with that of understanding the spectrum of various Hermitian matrices. The two identities that realise this trick are, firstly, Jensen’s formula

that relates the local distribution of eigenvalues to the log-determinants , and secondly the elementary identity

that relates the log-determinants of to the log-determinants of the Hermitian matrices

The main difficulty is then to obtain concentration and universality results for the Hermitian log-determinants . This turns out to be a task that is analogous to the task of obtaining concentration for Wigner matrices (as we did in this recent paper), as well as central limit theorems for log-determinants of Wigner matrices (as we did in this other recent paper). In both of these papers, the main idea was to use the Four Moment Theorem for Wigner matrices (which can now be proven relatively easily by a combination of the local semi-circular law and resolvent swapping methods), combined with (in the latter paper) a central limit theorem for the gaussian unitary ensemble (GUE). This latter task was achieved by using the convenient Trotter normal form to tridiagonalise a GUE matrix, which has the effect of revealing the determinant of that matrix as the solution to a certain linear stochastic difference equation, and one can analyse the distribution of that solution via such tools as the martingale central limit theorem.

The matrices are somewhat more complicated than Wigner matrices (for instance, the semi-circular law must be replaced by a distorted Marchenko-Pastur law), but the same general strategy works to obtain concentration and universality for their log-determinants. The main new difficulty that arises is that the analogue of the Trotter norm for gaussian random matrices is not tridiagonal, but rather Hessenberg (i.e. upper-triangular except for the lower diagonal). This ultimately has the effect of expressing the relevant determinant as the solution to a *nonlinear* stochastic difference equation, which is a bit trickier to solve for. Fortunately, it turns out that one only needs good lower bounds on the solution, as one can use the second moment method to upper bound the determinant and hence the log-determinant (following a classical computation of Turan). This simplifies the analysis on the equation somewhat.

While this result is the first local universality result in the category of random matrices with independent entries, there are still two limitations to the result which one would like to remove. The first is the moment matching hypotheses on the matrix. Very recently, one of the ingredients of our paper, namely the local circular law, was proved without moment matching hypotheses by Bourgade, Yau, and Yin (provided one stays away from the edge of the spectrum); however, as of this time of writing the other main ingredient – the universality of the log-determinant – still requires moment matching. (The standard tool for obtaining universality without moment matching hypotheses is the heat flow method (and more specifically, the local relaxation flow method), but the analogue of Dyson Brownian motion in the non-Hermitian setting appears to be somewhat intractible, being a coupled flow on both the eigenvalues and eigenvectors rather than just on the eigenvalues alone.)

Ben Green and I have just uploaded to the arXiv our paper “New bounds for Szemeredi’s theorem, Ia: Progressions of length 4 in finite field geometries revisited“, submitted to Proc. Lond. Math. Soc.. This is both an erratum to, and a replacement for, our previous paper “New bounds for Szemeredi’s theorem. I. Progressions of length 4 in finite field geometries“. The main objective in both papers is to bound the quantity for a vector space over a finite field of characteristic greater than , where is defined as the cardinality of the largest subset of that does not contain an arithmetic progression of length . In our earlier paper, we gave two arguments that bounded in the regime when the field was fixed and was large. The first “cheap” argument gave the bound

and the more complicated “expensive” argument gave the improvement

for some constant depending only on .

Unfortunately, while the cheap argument is correct, we discovered a subtle but serious gap in our expensive argument in the original paper. Roughly speaking, the strategy in that argument is to employ the *density increment method*: one begins with a large subset of that has no arithmetic progressions of length , and seeks to locate a subspace on which has a significantly increased density. Then, by using a “Koopman-von Neumann theorem”, ultimately based on an iteration of the inverse theorem of Ben and myself (and also independently by Samorodnitsky), one approximates by a “quadratically structured” function , which is (locally) a combination of a bounded number of quadratic phase functions, which one can prepare to be in a certain “locally equidistributed” or “locally high rank” form. (It is this reduction to the high rank case that distinguishes the “expensive” argument from the “cheap” one.) Because has no progressions of length , the count of progressions of length weighted by will also be small; by combining this with the theory of equidistribution of quadratic phase functions, one can then conclude that there will be a subspace on which has increased density.

The error in the paper was to conclude from this that the original function also had increased density on the same subspace; it turns out that the manner in which approximates is not strong enough to deduce this latter conclusion from the former. (One can strengthen the nature of approximation until one restores such a conclusion, but only at the price of deteriorating the quantitative bounds on one gets at the end of the day to be worse than the cheap argument.)

After trying unsuccessfully to repair this error, we eventually found an alternate argument, based on earlier papers of ourselves and of Bergelson-Host-Kra, that avoided the density increment method entirely and ended up giving a simpler proof of a stronger result than (1), and also gives the explicit value of for the exponent in (1). In fact, it gives the following stronger result:

Theorem 1Let be a subset of of density at least , and let . Then there is a subspace of of codimension such that the number of (possibly degenerate) progressions in is at least .

The bound (1) is an easy consequence of this theorem after choosing and removing the degenerate progressions from the conclusion of the theorem.

The main new idea is to work with a *local* Koopman-von Neumann theorem rather than a global one, trading a relatively weak global approximation to with a significantly stronger local approximation to on a subspace . This is somewhat analogous to how sometimes in graph theory it is more efficient (from the point of view of quantative estimates) to work with a local version of the Szemerédi regularity lemma which gives just a single regular pair of cells, rather than attempting to regularise almost all of the cells. This local approach is well adapted to the inverse theorem we use (which also has this local aspect), and also makes the reduction to the high rank case much cleaner. At the end of the day, one ends up with a fairly large subspace on which is quite dense (of density ) and which can be well approximated by a “pure quadratic” object, namely a function of a small number of quadratic phases obeying a high rank condition. One can then exploit a special positivity property of the count of length four progressions weighted by pure quadratic objects, essentially due to Bergelson-Host-Kra, which then gives the required lower bound.

## Recent Comments