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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).

Perhaps the most important structural result about general large dense graphs is the Szemerédi regularity lemma. Here is a standard formulation of that lemma:

Lemma 1 (Szemerédi regularity lemma)Let be a graph on vertices, and let . Then there exists a partition for some with the property that for all but at most of the pairs , the pair is-regularin the sense thatwhenever are such that and , and is the edge density between and . Furthermore, the partition is

equitablein the sense that for all .

There are many proofs of this lemma, which is actually not that difficult to establish; see for instance these previous blog posts for some examples. In this post I would like to record one further proof, based on the spectral decomposition of the adjacency matrix of , which is essentially due to Frieze and Kannan. (Strictly speaking, Frieze and Kannan used a variant of this argument to establish a weaker form of the regularity lemma, but it is not difficult to modify the Frieze-Kannan argument to obtain the usual form of the regularity lemma instead. Some closely related spectral regularity lemmas were also developed by Szegedy.) I found recently (while speaking at the Abel conference in honour of this year’s laureate, Endre Szemerédi) that this particular argument is not as widely known among graph theory experts as I had thought, so I thought I would record it here.

For reasons of exposition, it is convenient to first establish a slightly weaker form of the lemma, in which one drops the hypothesis of equitability (but then has to weight the cells by their magnitude when counting bad pairs):

Lemma 2 (Szemerédi regularity lemma, weakened variant). Let be a graph on vertices, and let . Then there exists a partition for some with the property that for all pairs outside of an exceptional set , one haswhenever , for some real number , where is the number of edges between and . Furthermore, we have

Let us now prove Lemma 2. We enumerate (after relabeling) as . The adjacency matrix of the graph is then a self-adjoint matrix, and thus admits an eigenvalue decomposition

for some orthonormal basis of and some eigenvalues , which we arrange in decreasing order of magnitude:

We can compute the trace of as

Among other things, this implies that

Let be a function (depending on ) to be chosen later, with for all . Applying (3) and the pigeonhole principle (or the finite convergence principle, see this blog post), we can find such that

(Indeed, the bound on is basically iterated times.) We can now split

where is the “structured” component

and is the “pseudorandom” component

We now design a vertex partition to make approximately constant on most cells. For each , we partition into cells on which (viewed as a function from to ) only fluctuates by , plus an exceptional cell of size coming from the values where is excessively large (larger than ). Combining all these partitions together, we can write for some , where has cardinality at most , and for all , the eigenfunctions all fluctuate by at most . In particular, if , then (by (4) and (6)) the entries of fluctuate by at most on each block . If we let be the mean value of these entries on , we thus have

for any and , where we view the indicator functions as column vectors of dimension .

Next, we observe from (3) and (7) that . If we let be the coefficients of , we thus have

and hence by Markov’s inequality we have

for all pairs outside of an exceptional set with

for any , by (10) and the Cauchy-Schwarz inequality.

Finally, to control we see from (4) and (8) that has an operator norm of at most . In particular, we have from the Cauchy-Schwarz inequality that

Let be the set of all pairs where either , , , or

One easily verifies that (2) holds. If is not in , then by summing (9), (11), (12) and using (5), we see that

for all . The left-hand side is just . As , we have

and so (since )

If we let be a sufficiently rapidly growing function of that depends on , the second error term in (13) can be absorbed in the first, and (1) follows. This concludes the proof of Lemma 2.

To prove Lemma 1, one argues similarly (after modifying as necessary), except that the initial partition of constructed above needs to be subdivided further into equitable components (of size ), plus some remainder sets which can be aggregated into an exceptional component of size (and which can then be redistributed amongst the other components to arrive at a truly equitable partition). We omit the details.

Remark 1It is easy to verify that needs to be growing exponentially in in order for the above argument to work, which leads to tower-exponential bounds in the number of cells in the partition. It was shown by Gowers that a tower-exponential bound is actually necessary here. By varying , one basically obtains thestrong regularity lemmafirst established by Alon, Fischer, Krivelevich, and Szegedy; in the opposite direction, setting essentially gives theweak regularity lemmaof Frieze and Kannan.

Remark 2If we specialise to a Cayley graph, in which is a finite abelian group and for some (symmetric) subset of , then the eigenvectors are characters, and one essentially recovers thearithmetic regularity lemmaof Green, in which the vertex partition classes are given by Bohr sets (and one can then place additional regularity properties on these Bohr sets with some additional arguments). The components of , representing high, medium, and low eigenvalues of , then become a decomposition associated to high, medium, and low Fourier coefficients of .

Remark 3The use of spectral theory here is parallel to the use of Fourier analysis to establish results such as Roth’s theorem on arithmetic progressions of length three. In analogy with this, one could view hypergraph regularity as being a sort of “higher order spectral theory”, although this spectral perspective is not as convenient as it is in the graph case.

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.

One of the first non-trivial theorems one encounters in classical algebraic geometry is Bézout’s theorem, which we will phrase as follows:

Theorem 1 (Bézout’s theorem)Let be a field, and let be non-zero polynomials in two variables with no common factor. Then the two curves and have no common components, and intersect in at most points.

This theorem can be proven by a number of means, for instance by using the classical tool of resultants. It has many strengthenings, generalisations, and variants; see for instance this previous blog post on Bézout’s inequality. Bézout’s theorem asserts a fundamental algebraic dichotomy, of importance in combinatorial incidence geometry: any two algebraic curves either share a common component, or else have a bounded finite intersection; there is no intermediate case in which the intersection is unbounded in cardinality, but falls short of a common component. This dichotomy is closely related to the integrality gap in algebraic dimension: an algebraic set can have an integer dimension such as or , but cannot attain any intermediate dimensions such as . This stands in marked contrast to sets of analytic, combinatorial, or probabilistic origin, whose “dimension” is typically not necessarily constrained to be an integer.

Bézout’s inequality tells us, roughly speaking, that the intersection of a curve of degree and a curve of degree forms a set of at most points. One can consider the converse question: given a set of points in the plane , can one find two curves of degrees with and no common components, whose intersection contains these points?

A model example that supports the possibility of such a converse is a grid that is a Cartesian product of two finite subsets of with . In this case, one can take one curve to be the union of vertical lines, and the other curve to be the union of horizontal lines, to obtain the required decomposition. Thus, if the proposed converse to Bézout’s inequality held, it would assert that any set of points was essentially behaving like a “nonlinear grid” of size .

Unfortunately, the naive converse to Bézout’s theorem is false. A counterexample can be given by considering a set of points for some large perfect square , where is a by grid of the form described above, and consists of points on an line (e.g. a or grid). Each of the two component sets can be written as the intersection between two curves whose degrees multiply up to ; in the case of , we can take the two families of parallel lines (viewed as reducible curves of degree ) as the curves, and in the case of , one can take as one curve, and the graph of a degree polynomial on vanishing on for the second curve. But, if is large enough, one cannot cover by the intersection of a single pair of curves with no common components whose degrees multiply up to . Indeed, if this were the case, then without loss of generality we may assume that , so that . By Bézout’s theorem, either contains , or intersects in at most points. Thus, in order for to capture all of , it must contain , which forces to not contain . But has to intersect in points, so by Bézout’s theorem again we have , thus . But then (by more applications of Bézout’s theorem) can only capture of the points of , a contradiction.

But the above counterexample suggests that even if an arbitrary set of (or ) points cannot be covered by the single intersection of a pair of curves with degree multiplying up to , one may be able to cover such a set by a small number of such intersections. The purpose of this post is to record the simple observation that this is, indeed, the case:

Theorem 2 (Partial converse to Bézout’s theorem)Let be a field, and let be a set of points in for some . Then one can find and pairs of coprime non-zero polynomials for such that

Informally, every finite set in the plane is (a dense subset of) the union of logarithmically many nonlinear grids. The presence of the logarithm is necessary, as can be seen by modifying the example to be the union of logarithmically many Cartesian products of distinct dimensions, rather than just a pair of such products.

Unfortunately I do not know of any application of this converse, but I thought it was cute anyways. The proof is given below the fold.

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.

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.

A few days ago, Endre Szemerédi was awarded the 2012 Abel prize “for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.” The full citation for the prize may be found here, and the written notes for a talk given by Tim Gowers on Endre’s work at the announcement may be found here (and video of the talk can be found here).

As I was on the Abel prize committee this year, I won’t comment further on the prize, but will instead focus on what is arguably Endre’s most well known result, namely Szemerédi’s theorem on arithmetic progressions:

Theorem 1 (Szemerédi’s theorem)Let be a set of integers of positive upper density, thus , where . Then contains an arithmetic progression of length for any .

Szemerédi’s original proof of this theorem is a remarkably intricate piece of combinatorial reasoning. Most proofs of theorems in mathematics – even long and difficult ones – generally come with a reasonably compact “high-level” overview, in which the proof is (conceptually, at least) broken down into simpler pieces. There may well be technical difficulties in formulating and then proving each of the component pieces, and then in fitting the pieces together, but usually the “big picture” is reasonably clear. To give just one example, the overall strategy of Perelman’s proof of the Poincaré conjecture can be briefly summarised as follows: to show that a simply connected three-dimensional manifold is homeomorphic to a sphere, place a Riemannian metric on it and perform Ricci flow, excising any singularities that arise by surgery, until the entire manifold becomes extinct. By reversing the flow and analysing the surgeries performed, obtain enough control on the topology of the original manifold to establish that it is a topological sphere.

In contrast, the pieces of Szemerédi’s proof are highly interlocking, particularly with regard to all the epsilon-type parameters involved; it takes quite a bit of notational setup and foundational lemmas before the key steps of the proof can even be stated, let alone proved. Szemerédi’s original paper contains a logical diagram of the proof (reproduced in Gowers’ recent talk) which already gives a fair indication of this interlocking structure. (Many years ago I tried to present the proof, but I was unable to find much of a simplification, and my exposition is probably not that much clearer than the original text.) Even the use of nonstandard analysis, which is often helpful in cleaning up armies of epsilons, turns out to be a bit tricky to apply here. (In typical applications of nonstandard analysis, one can get by with a single nonstandard universe, constructed as an ultrapower of the standard universe; but to correctly model all the epsilons occuring in Szemerédi’s argument, one needs to repeatedly perform the ultrapower construction to obtain a (finite) sequence of increasingly nonstandard (and increasingly saturated) universes, each one containing unbounded quantities that are far larger than any quantity that appears in the preceding universe, as discussed at the end of this previous blog post. This sequence of universes does end up concealing all the epsilons, but it is not so clear that this is a net gain in clarity for the proof; I may return to the nonstandard presentation of Szemeredi’s argument at some future juncture.)

Instead of trying to describe the entire argument here, I thought I would instead show some key components of it, with only the slightest hint as to how to assemble the components together to form the whole proof. In particular, I would like to show how two particular ingredients in the proof – namely van der Waerden’s theorem and the Szemerédi regularity lemma – become useful. For reasons that will hopefully become clearer later, it is convenient not only to work with ordinary progressions , but also progressions of progressions , progressions of progressions of progressions, and so forth. (In additive combinatorics, these objects are known as *generalised arithmetic progressions* of rank one, two, three, etc., and play a central role in the subject, although the way they are used in Szemerédi’s proof is somewhat different from the way that they are normally used in additive combinatorics.) Very roughly speaking, Szemerédi’s proof begins by building an enormous generalised arithmetic progression of high rank containing many elements of the set (arranged in a “near-maximal-density” configuration), and then steadily prunes this progression to improve the combinatorial properties of the configuration, until one ends up with a single rank one progression of length that consists entirely of elements of .

To illustrate some of the basic ideas, let us first consider a situation in which we have located a progression of progressions of length , with each progression , being quite long, and containing a near-maximal amount of elements of , thus

where is the “maximal density” of along arithmetic progressions. (There are a lot of subtleties in the argument about exactly how good the error terms are in various approximations, but we will ignore these issues for the sake of this discussion and just use the imprecise symbols such as instead.) By hypothesis, is positive. The objective is then to locate a progression in , with each in for . It may help to view the progression of progressions as a tall thin rectangle .

If we write for , then the problem is equivalent to finding a (possibly degenerate) arithmetic progression , with each in .

By hypothesis, we know already that each set has density about in :

Let us now make a “weakly mixing” assumption on the , which roughly speaking asserts that

for “most” subsets of of density of a certain form to be specified shortly. This is a plausible type of assumption if one believes to behave like a random set, and if the sets are constructed “independently” of the in some sense. Of course, we do not expect such an assumption to be valid all of the time, but we will postpone consideration of this point until later. Let us now see how this sort of weakly mixing hypothesis could help one count progressions of the desired form.

We will inductively consider the following (nonrigorously defined) sequence of claims for each :

- : For most choices of , there are arithmetic progressions in with the specified choice of , such that for all .

(Actually, to avoid boundary issues one should restrict to lie in the middle third of , rather than near the edges, but let us ignore this minor technical detail.) The quantity is natural here, given that there are arithmetic progressions in that pass through in the position, and that each one ought to have a probability of or so that the events simultaneously hold.) If one has the claim , then by selecting a typical in , we obtain a progression with for all , as required. (In fact, we obtain about such progressions by this method.)

We can heuristically justify the claims by induction on . For , the claims are clear just from direct counting of progressions (as long as we keep away from the edges of ). Now suppose that , and the claims have already been proven. For any and for most , we have from hypothesis that there are progressions in through with . Let be the set of all the values of attained by these progressions, then . Invoking the weak mixing hypothesis, we (heuristically, at least) conclude that for most choices of , we have

which then gives the desired claim .

The observant reader will note that we only needed the claim in the case for the above argument, but for technical reasons, the full proof requires one to work with more general values of (also the claim needs to be replaced by a more complicated version of itself, but let’s ignore this for sake of discussion).

We now return to the question of how to justify the weak mixing hypothesis (2). For a single block of , one can easily concoct a scenario in which this hypothesis fails, by choosing to overlap with too strongly, or to be too disjoint from . However, one can do better if one can select from a long progression of blocks. The starting point is the following simple double counting observation that gives the right upper bound:

Proposition 2 (Single upper bound)Let be a progression of progressions for some large . Suppose that for each , the set has density in (i.e. (1) holds). Let be a subset of of density . Then (if is large enough) one can find an such that

*Proof:* The key is the double counting identity

Because has maximal density and is large, we have

for each , and thus

The claim then follows from the pigeonhole principle.

Now suppose we want to obtain weak mixing not just for a single set , but for a small number of such sets, i.e. we wish to find an for which

for all , where is the density of in . The above proposition gives, for each , a choice of for which (3) holds, but it could be a different for each , and so it is not immediately obvious how to use Proposition 2 to find an for which (3) holds *simultaneously* for all . However, it turns out that the van der Waerden theorem is the perfect tool for this amplification:

Proposition 3 (Multiple upper bound)Let be a progression of progressions for some large . Suppose that for each , the set has density in (i.e. (1) holds). For each , let be a subset of of density . Then (if is large enough depending on ) one can find an such thatsimultaneously for all .

*Proof:* Suppose that the claim failed (for some suitably large ). Then, for each , there exists such that

This can be viewed as a colouring of the interval by colours. If we take large compared to , van der Waerden’s theorem allows us to then find a long subprogression of which is monochromatic, so that is constant on this progression. But then this will furnish a counterexample to Proposition 2.

One nice thing about this proposition is that the upper bounds can be automatically upgraded to an asymptotic:

Proposition 4 (Multiple mixing)Let be a progression of progressions for some large . Suppose that for each , the set has density in (i.e. (1) holds). For each , let be a subset of of density . Then (if is large enough depending on ) one can find an such thatsimultaneously for all .

*Proof:* By applying the previous proposition to the collection of sets and their complements (thus replacing with , one can find an for which

and

which gives the claim.

However, this improvement of Proposition 2 turns out to not be strong enough for applications. The reason is that the number of sets for which mixing is established is too small compared with the length of the progression one has to use in order to obtain that mixing. However, thanks to the magic of the Szemerédi regularity lemma, one can amplify the above proposition even further, to allow for a huge number of to be mixed (at the cost of excluding a small fraction of exceptions):

Proposition 5 (Really multiple mixing)Let be a progression of progressions for some large . Suppose that for each , the set has density in (i.e. (1) holds). For each in some (large) finite set , let be a subset of of density . Then (if is large enough, butnotdependent on the size of ) one can find an such thatsimultaneously for almost all .

*Proof:* We build a bipartite graph connecting the progression to the finite set by placing an edge between an element and an element whenever . The number can then be interpreted as the degree of in this graph, while the number is the number of neighbours of that land in .

We now apply the regularity lemma to this graph . Roughly speaking, what this lemma does is to partition and into almost equally sized cells and such that for most pairs of cells, the graph resembles a random bipartite graph of some density between these two cells. The key point is that the number of cells here is bounded uniformly in the size of and . As a consequence of this lemma, one can show that for most vertices in a typical cell , the number is approximately equal to

and the number is approximately equal to

The point here is that the different statistics are now controlled by a mere statistics (this is not unlike the use of principal component analysis in statistics, incidentally, but that is another story). Now, we invoke Proposition 4 to find an for which

simultaneously for all , and the claim follows.

This proposition now suggests a way forward to establish the type of mixing properties (2) needed for the preceding attempt at proving Szemerédi’s theorem to actually work. Whereas in that attempt, we were working with a single progression of progressions of progressions containing a near-maximal density of elements of , we will now have to work with a *family* of such progression of progressions, where ranges over some suitably large parameter set. Furthermore, in order to invoke Proposition 5, this family must be “well-arranged” in some arithmetic sense; in particular, for a given , it should be possible to find many reasonably large subfamilies of this family for which the terms of the progression of progressions in this subfamily are themselves in arithmetic progression. (Also, for technical reasons having to do with the fact that the sets in Proposition 5 are not allowed to depend on , one also needs the progressions for any given to be “similar” in the sense that they intersect in the same fashion (thus the sets as varies need to be translates of each other).) If one has this sort of family, then Proposition 5 allows us to “spend” some of the degrees of freedom of the parameter set in order to gain good mixing properties for at least one of the sets in the progression of progressions.

Of course, we still have to figure out how to get such large families of well-arranged progressions of progressions. Szemerédi’s solution was to begin by working with generalised progressions of a much larger rank than the rank progressions considered here; roughly speaking, to prove Szemerédi’s theorem for length progressions, one has to consider generalised progressions of rank as high as . It is possible by a reasonably straightforward (though somewhat delicate) “density increment argument” to locate a huge generalised progression of this rank which is “saturated” by in a certain rather technical sense (related to the concept of “near maximal density” used previously). Then, by another reasonably elementary argument, it is possible to locate inside a suitable large generalised progression of some rank , a family of large generalised progressions of rank which inherit many of the good properties of the original generalised progression, and which have the arithmetic structure needed for Proposition 5 to be applicable, at least for one value of . (But getting this sort of property for *all* values of simultaneously is tricky, and requires many careful iterations of the above scheme; there is also the problem that by obtaining good behaviour for one index , one may lose good behaviour at previous indices, leading to a sort of “Tower of Hanoi” situation which may help explain the exponential factor in the rank that is ultimately needed. It is an extremely delicate argument; all the parameters and definitions have to be set very precisely in order for the argument to work at all, and it is really quite remarkable that Endre was able to see it through to the end.)

This is an addendum to last quarter’s course notes on Hilbert’s fifth problem, which I am in the process of reviewing in order to transcribe them into a book (as was done similarly for several other sets of lecture notes on this blog). When reviewing the zeroth set of notes in particular, I found that I had made a claim (Proposition 11 from those notes) which asserted, roughly speaking, that any sufficiently large nilprogression was an approximate group, and promised to prove it later in the course when we had developed the ability to calculate efficiently in nilpotent groups. As it turned out, I managed finish the course without the need to develop these calculations, and so the proposition remained unproven. In order to rectify this, I will use this post to lay out some of the basic algebra of nilpotent groups, and use it to prove the above proposition, which turns out to be a bit tricky. (In my paper with Breuillard and Green, we avoid the need for this proposition by restricting attention to a special type of nilprogression, which we call a nilprogression in -normal form, for which the computations are simpler.)

There are several ways to think about nilpotent groups; for instance one can use the model example of the Heisenberg group

over an arbitrary ring (which need not be commutative), or more generally any matrix group consisting of unipotent upper triangular matrices, and view a general nilpotent group as being an abstract generalisation of such concrete groups. (In the case of nilpotent Lie groups, at least, this is quite an accurate intuition, thanks to Engel’s theorem.) Or, one can adopt a Lie-theoretic viewpoint and try to think of nilpotent groups as somehow arising from nilpotent Lie algebras; this intuition is rigorous when working with nilpotent Lie groups (at least when the characteristic is large, in order to avoid issues coming from the denominators in the Baker-Campbell-Hausdorff formula), but also retains some conceptual value in the non-Lie setting. In particular, nilpotent groups (particularly finitely generated ones) can be viewed in some sense as “nilpotent Lie groups over “, even though Lie theory does not quite work perfectly when the underlying scalars merely form an integral domain instead of a field.

Another point of view, which arises naturally both in analysis and in algebraic geometry, is to view nilpotent groups as modeling “infinitesimal” perturbations of the identity, where the infinitesimals have a certain finite order. For instance, given a (not necessarily commutative) ring without identity (representing all the “small” elements of some larger ring or algebra), we can form the powers for , defined as the ring generated by -fold products of elements in ; this is an ideal of which represents the elements which are “ order” in some sense. If one then formally adjoins an identity onto the ring , then for any , the multiplicative group is a nilpotent group of step at most . For instance, if is the ring of strictly upper matrices (over some base ring), then vanishes and becomes the group of unipotent upper triangular matrices over the same ring, thus recovering the previous matrix-based example. In analysis applications, might be a ring of operators which are somehow of “order” or for some small parameter or , and one wishes to perform Taylor expansions up to order or , thus discarding (i.e. quotienting out) all errors in .

From a dynamical or group-theoretic perspective, one can also view nilpotent groups as towers of central extensions of a trivial group. Finitely generated nilpotent groups can also be profitably viewed as a special type of polycylic group; this is the perspective taken in this previous blog post. Last, but not least, one can view nilpotent groups from a combinatorial group theory perspective, as being words from some set of generators of various “degrees” subject to some commutation relations, with commutators of two low-degree generators being expressed in terms of higher degree objects, and all commutators of a sufficiently high degree vanishing. In particular, generators of a given degree can be moved freely around a word, as long as one is willing to generate commutator errors of higher degree.

With this last perspective, in particular, one can start computing in nilpotent groups by adopting the philosophy that the lowest order terms should be attended to first, without much initial concern for the higher order errors generated in the process of organising the lower order terms. Only after the lower order terms are in place should attention then turn to higher order terms, working successively up the hierarchy of degrees until all terms are dealt with. This turns out to be a relatively straightforward philosophy to implement in many cases (particularly if one is not interested in explicit expressions and constants, being content instead with qualitative expansions of controlled complexity), but the arguments are necessarily recursive in nature and as such can become a bit messy, and require a fair amount of notation to express precisely. So, unfortunately, the arguments here will be somewhat cumbersome and notation-heavy, even if the underlying methods of proof are relatively simple.

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