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(This is an extended blog post version of my talk “Ultraproducts as a Bridge Between Discrete and Continuous Analysis” that I gave at the Simons institute for the theory of computing at the workshop “Neo-Classical methods in discrete analysis“. Some of the material here is drawn from previous blog posts, notably “Ultraproducts as a bridge between hard analysis and soft analysis” and “Ultralimit analysis and quantitative algebraic geometry“‘. The text here has substantially more details than the talk; one may wish to skip all of the proofs given here to obtain a closer approximation to the original talk.)
Discrete analysis, of course, is primarily interested in the study of discrete (or “finitary”) mathematical objects: integers, rational numbers (which can be viewed as ratios of integers), finite sets, finite graphs, finite or discrete metric spaces, and so forth. However, many powerful tools in mathematics (e.g. ergodic theory, measure theory, topological group theory, algebraic geometry, spectral theory, etc.) work best when applied to continuous (or “infinitary”) mathematical objects: real or complex numbers, manifolds, algebraic varieties, continuous topological or metric spaces, etc. In order to apply results and ideas from continuous mathematics to discrete settings, there are basically two approaches. One is to directly discretise the arguments used in continuous mathematics, which often requires one to keep careful track of all the bounds on various quantities of interest, particularly with regard to various error terms arising from discretisation which would otherwise have been negligible in the continuous setting. The other is to construct continuous objects as limits of sequences of discrete objects of interest, so that results from continuous mathematics may be applied (often as a “black box”) to the continuous limit, which then can be used to deduce consequences for the original discrete objects which are quantitative (though often ineffectively so). The latter approach is the focus of this current talk.
The following table gives some examples of a discrete theory and its continuous counterpart, together with a limiting procedure that might be used to pass from the former to the latter:
(Discrete) | (Continuous) | (Limit method) |
Ramsey theory | Topological dynamics | Compactness |
Density Ramsey theory | Ergodic theory | Furstenberg correspondence principle |
Graph/hypergraph regularity | Measure theory | Graph limits |
Polynomial regularity | Linear algebra | Ultralimits |
Structural decompositions | Hilbert space geometry | Ultralimits |
Fourier analysis | Spectral theory | Direct and inverse limits |
Quantitative algebraic geometry | Algebraic geometry | Schemes |
Discrete metric spaces | Continuous metric spaces | Gromov-Hausdorff limits |
Approximate group theory | Topological group theory | Model theory |
As the above table illustrates, there are a variety of different ways to form a limiting continuous object. Roughly speaking, one can divide limits into three categories:
- Topological and metric limits. These notions of limits are commonly used by analysts. Here, one starts with a sequence (or perhaps a net) of objects in a common space , which one then endows with the structure of a topological space or a metric space, by defining a notion of distance between two points of the space, or a notion of open neighbourhoods or open sets in the space. Provided that the sequence or net is convergent, this produces a limit object , which remains in the same space, and is “close” to many of the original objects with respect to the given metric or topology.
- Categorical limits. These notions of limits are commonly used by algebraists. Here, one starts with a sequence (or more generally, a diagram) of objects in a category , which are connected to each other by various morphisms. If the ambient category is well-behaved, one can then form the direct limit or the inverse limit of these objects, which is another object in the same category , and is connected to the original objects by various morphisms.
- Logical limits. These notions of limits are commonly used by model theorists. Here, one starts with a sequence of objects or of spaces , each of which is (a component of) a model for given (first-order) mathematical language (e.g. if one is working in the language of groups, might be groups and might be elements of these groups). By using devices such as the ultraproduct construction, or the compactness theorem in logic, one can then create a new object or a new space , which is still a model of the same language (e.g. if the spaces were all groups, then the limiting space will also be a group), and is “close” to the original objects or spaces in the sense that any assertion (in the given language) that is true for the limiting object or space, will also be true for many of the original objects or spaces, and conversely. (For instance, if is an abelian group, then the will also be abelian groups for many .)
The purpose of this talk is to highlight the third type of limit, and specifically the ultraproduct construction, as being a “universal” limiting procedure that can be used to replace most of the limits previously mentioned. Unlike the topological or metric limits, one does not need the original objects to all lie in a common space in order to form an ultralimit ; they are permitted to lie in different spaces ; this is more natural in many discrete contexts, e.g. when considering graphs on vertices in the limit when goes to infinity. Also, no convergence properties on the are required in order for the ultralimit to exist. Similarly, ultraproduct limits differ from categorical limits in that no morphisms between the various spaces involved are required in order to construct the ultraproduct.
With so few requirements on the objects or spaces , the ultraproduct construction is necessarily a very “soft” one. Nevertheless, the construction has two very useful properties which make it particularly useful for the purpose of extracting good continuous limit objects out of a sequence of discrete objects. First of all, there is Łos’s theorem, which roughly speaking asserts that any first-order sentence which is asymptotically obeyed by the , will be exactly obeyed by the limit object ; in particular, one can often take a discrete sequence of “partial counterexamples” to some assertion, and produce a continuous “complete counterexample” that same assertion via an ultraproduct construction; taking the contrapositives, one can often then establish a rigorous equivalence between a quantitative discrete statement and its qualitative continuous counterpart. Secondly, there is the countable saturation property that ultraproducts automatically enjoy, which is a property closely analogous to that of compactness in topological spaces, and can often be used to ensure that the continuous objects produced by ultraproduct methods are “complete” or “compact” in various senses, which is particularly useful in being able to upgrade qualitative (or “pointwise”) bounds to quantitative (or “uniform”) bounds, more or less “for free”, thus reducing significantly the burden of “epsilon management” (although the price one pays for this is that one needs to pay attention to which mathematical objects of study are “standard” and which are “nonstandard”). To achieve this compactness or completeness, one sometimes has to restrict to the “bounded” portion of the ultraproduct, and it is often also convenient to quotient out the “infinitesimal” portion in order to complement these compactness properties with a matching “Hausdorff” property, thus creating familiar examples of continuous spaces, such as locally compact Hausdorff spaces.
Ultraproducts are not the only logical limit in the model theorist’s toolbox, but they are one of the simplest to set up and use, and already suffice for many of the applications of logical limits outside of model theory. In this post, I will set out the basic theory of these ultraproducts, and illustrate how they can be used to pass between discrete and continuous theories in each of the examples listed in the above table.
Apart from the initial “one-time cost” of setting up the ultraproduct machinery, the main loss one incurs when using ultraproduct methods is that it becomes very difficult to extract explicit quantitative bounds from results that are proven by transferring qualitative continuous results to the discrete setting via ultraproducts. However, in many cases (particularly those involving regularity-type lemmas) the bounds are already of tower-exponential type or worse, and there is arguably not much to be lost by abandoning the explicit quantitative bounds altogether.
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 -regular in the sense that
whenever are such that and , and is the edge density between and . Furthermore, the partition is equitable in 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 has
whenever , 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 1 It 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 the strong regularity lemma first established by Alon, Fischer, Krivelevich, and Szegedy; in the opposite direction, setting essentially gives the weak regularity lemma of Frieze and Kannan.
Remark 2 If 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 the arithmetic regularity lemma of 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 3 The 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.
Ben Green, and I have just uploaded to the arXiv our paper “An arithmetic regularity lemma, an associated counting lemma, and applications“, submitted (a little behind schedule) to the 70th birthday conference proceedings for Endre Szemerédi. In this paper we describe the general-degree version of the arithmetic regularity lemma, which can be viewed as the counterpart of the Szemerédi regularity lemma, in which the object being regularised is a function on a discrete interval rather than a graph, and the type of patterns one wishes to count are additive patterns (such as arithmetic progressions ) rather than subgraphs. Very roughly speaking, this regularity lemma asserts that all such functions can be decomposed as a degree nilsequence (or more precisely, a variant of a nilsequence that we call an virtual irrational nilsequence), plus a small error, plus a third error which is extremely tiny in the Gowers uniformity norm . In principle, at least, the latter two errors can be readily discarded in applications, so that the regularity lemma reduces many questions in additive combinatorics to questions concerning (virtual irrational) nilsequences. To work with these nilsequences, we also establish a arithmetic counting lemma that gives an integral formula for counting additive patterns weighted by such nilsequences.
The regularity lemma is a manifestation of the “dichotomy between structure and randomness”, as discussed for instance in my ICM article or FOCS article. In the degree case , this result is essentially due to Green. It is powered by the inverse conjecture for the Gowers norms, which we and Tamar Ziegler have recently established (paper to be forthcoming shortly; the case of our argument is discussed here). The counting lemma is established through the quantitative equidistribution theory of nilmanifolds, which Ben and I set out in this paper.
The regularity and counting lemmas are designed to be used together, and in the paper we give three applications of this combination. Firstly, we give a new proof of Szemerédi’s theorem, which proceeds via an energy increment argument rather than a density increment one. Secondly, we establish a conjecture of Bergelson, Host, and Kra, namely that if has density , and , then there exist shifts for which contains at least arithmetic progressions of length of spacing . (The case of this conjecture was established earlier by Green; the case is false, as was shown by Ruzsa in an appendix to the Bergelson-Host-Kra paper.) Thirdly, we establish a variant of a recent result of Gowers-Wolf, showing that the true complexity of a system of linear forms over indeed matches the conjectured value predicted in their first paper.
In all three applications, the scheme of proof can be described as follows:
- Apply the arithmetic regularity lemma, and decompose a relevant function into three pieces, .
- The uniform part is so tiny in the Gowers uniformity norm that its contribution can be easily dealt with by an appropriate “generalised von Neumann theorem”.
- The contribution of the (virtual, irrational) nilsequence can be controlled using the arithmetic counting lemma.
- Finally, one needs to check that the contribution of the small error does not overwhelm the main term . This is the trickiest bit; one often needs to use the counting lemma again to show that one can find a set of arithmetic patterns for that is so sufficiently “equidistributed” that it is not impacted by the small error.
To illustrate the last point, let us give the following example. Suppose we have a set of some positive density (say ) and we have managed to prove that contains a reasonable number of arithmetic progressions of length (say), e.g. it contains at least such progressions. Now we perturb by deleting a small number, say , elements from to create a new set . Can we still conclude that the new set contains any arithmetic progressions of length ?
Unfortunately, the answer could be no; conceivably, all of the arithmetic progressions in could be wiped out by the elements removed from , since each such element of could be associated with up to (or even ) arithmetic progressions in .
But suppose we knew that the arithmetic progressions in were equidistributed, in the sense that each element in belonged to the same number of such arithmetic progressions, namely . Then each element deleted from only removes at most progressions, and so one can safely remove elements from and still retain some arithmetic progressions. The same argument works if the arithmetic progressions are only approximately equidistributed, in the sense that the number of progressions that a given element belongs to concentrates sharply around its mean (for instance, by having a small variance), provided that the equidistribution is sufficiently strong. Fortunately, the arithmetic regularity and counting lemmas are designed to give precisely such a strong equidistribution result.
A succinct (but slightly inaccurate) summation of the regularity+counting lemma strategy would be that in order to solve a problem in additive combinatorics, it “suffices to check it for nilsequences”. But this should come with a caveat, due to the issue of the small error above; in addition to checking it for nilsequences, the answer in the nilsequence case must be sufficiently “dispersed” in a suitable sense, so that it can survive the addition of a small (but not completely negligible) perturbation.
One last “production note”. Like our previous paper with Emmanuel Breuillard, we used Subversion to write this paper, which turned out to be a significant efficiency boost as we could work on different parts of the paper simultaneously (this was particularly important this time round as the paper was somewhat lengthy and complicated, and there was a submission deadline). When doing so, we found it convenient to split the paper into a dozen or so pieces (one for each section of the paper, basically) in order to avoid conflicts, and to help coordinate the writing process. I’m also looking into git (a more advanced version control system), and am planning to use it for another of my joint projects; I hope to be able to comment on the relative strengths of these systems (and with plain old email) in the future.
This week I am at Rutgers University, giving the Lewis Memorial Lectures for this year, which are also concurrently part of a workshop in random matrices. I gave four lectures, three of which were on random matrices, and one of which was on the Szemerédi regularity lemma.
The titles, abstracts, and slides of these talks are as follows.
- Szemerédi’s lemma revisited. In this general-audience talk, I discuss the Szemerédi regularity lemma (which, roughly speaking, shows that an arbitrary large dense graph can always be viewed as the disjoint union of a bounded number of pseudorandom components), and how it has recently been reinterpreted in a more analytical (and infinitary) language using the theory of graph limits or of exchangeable measures. I also discuss arithmetic analogues of this lemma, including one which (implicitly) underlies my result with Ben Green that the primes contain arbitrarily long arithmetic progressions.
- Singularity and determinant of random matrices. Here, I present recent progress in understanding the question of how likely a random matrix (e.g. one whose entries are all +1 or -1 with equal probability) is to be invertible, as well as the related question of how large the determinant should be. The case of continuous matrix ensembles (such as the Gaussian ensemble) is well understood, but the discrete case contains some combinatorial difficulties and took longer to understand properly. In particular I present the results of Kahn-Komlós-Szemerédi and later authors showing that discrete random matrices are invertible with exponentially high probability, and also give some results for the distribution of the determinant.
- The least singular value of random matrices. A more quantitative version of the question “when is a matrix invertible?” is “what is the least singular value of that matrix”? I present here the recent results of Litvak-Pajor-Rudelson-Tomczak-Jaegermann, Rudelson, myself and Vu, and Rudelson–Vershynin on addressing this question in the discrete case. A central role is played by the inverse Littlewood-Offord theorems of additive combinatorics, which give reasonably sharp necessary conditions for a discrete random walk to concentrate in a small ball.
- The circular law. One interesting application of the above theory is to extend the circular law for the spectrum of random matrices from the continuous case to the discrete case. Previous arguments of Girko and Bai for the continuous case can be transplanted to the discrete case, but the key new ingredient needed is a least singular value bound for shifted matrices in order to avoid the spectrum being overwhelmed by pseudospectrum. It turns out that the results of the preceding lecture are almost precisely what are needed to accomplish this.
[Update, Mar 31: first lecture slides corrected. Thanks to Yoshiyasu Ishigami for pointing out a slight inaccuracy in the text.]
I’ve just uploaded to the arXiv my joint paper with Tim Austin, “On the testability and repair of hereditary hypergraph properties“, which has been submitted to Random Structures and Algorithms. In this paper we prove some positive and negative results for the testability (and the local repairability) of various properties of directed or undirected graphs and hypergraphs, which can be either monochromatic or multicoloured.
The negative results have already been discussed in a previous posting of mine, so today I will focus on the positive results. The property testing results here are finitary results, but it turns out to be rather convenient to use a certain correspondence principle (the hypergraph version of the Furstenberg correspondence principle) to convert the question into one about exchangeable probability measures on spaces of hypergraphs (i.e. on random hypergraphs whose probability distribution is invariant under exchange of vertices). Such objects are also closely related to the”graphons” and “hypergraphons” that emerge as graph limits, as studied by Lovasz-Szegedy, Elek-Szegedy, and others. Somewhat amusingly, once one does so, it then becomes convenient to keep track of objects indexed by vertex sets and how they are exchanged via the language of category theory, and in particular using the concept of a natural transformation to describe such objects as exchangeable measures, graph colourings, and local modification rules. I will try to sketch out some of these connections, after describing the main positive results.
This post is a sequel of sorts to my earlier post on hard and soft analysis, and the finite convergence principle. Here, I want to discuss a well-known theorem in infinitary soft analysis – the Lebesgue differentiation theorem – and whether there is any meaningful finitary version of this result. Along the way, it turns out that we will uncover a simple analogue of the Szemerédi regularity lemma, for subsets of the interval rather than for graphs. (Actually, regularity lemmas seem to appear in just about any context in which fine-scaled objects can be approximated by coarse-scaled ones.) The connection between regularity lemmas and results such as the Lebesgue differentiation theorem was recently highlighted by Elek and Szegedy, while the connection between the finite convergence principle and results such as the pointwise ergodic theorem (which is a close cousin of the Lebesgue differentiation theorem) was recently detailed by Avigad, Gerhardy, and Towsner.
The Lebesgue differentiation theorem has many formulations, but we will avoid the strongest versions and just stick to the following model case for simplicity:
Lebesgue differentiation theorem. If is Lebesgue measurable, then for almost every we have . Equivalently, the fundamental theorem of calculus is true for almost every x in [0,1].
Here we use the oriented definite integral, thus . Specialising to the case where is an indicator function, we obtain the Lebesgue density theorem as a corollary:
Lebesgue density theorem. Let be Lebesgue measurable. Then for almost every , we have as , where |A| denotes the Lebesgue measure of A.
In other words, almost all the points x of A are points of density of A, which roughly speaking means that as one passes to finer and finer scales, the immediate vicinity of x becomes increasingly saturated with A. (Points of density are like robust versions of interior points, thus the Lebesgue density theorem is an assertion that measurable sets are almost like open sets. This is Littlewood’s first principle.) One can also deduce the Lebesgue differentiation theorem back from the Lebesgue density theorem by approximating f by a finite linear combination of indicator functions; we leave this as an exercise.
In this second lecture, I wish to talk about the dichotomy between structure and randomness as it manifests itself in four closely related areas of mathematics:
- Combinatorial number theory, which seeks to find patterns in unstructured dense sets (or colourings) of integers;
- Ergodic theory (or more specifically, multiple recurrence theory), which seeks to find patterns in positive-measure sets under the action of a discrete dynamical system on probability spaces (or more specifically, measure-preserving actions of the integers );
- Graph theory, or more specifically the portion of this theory concerned with finding patterns in large unstructured dense graphs; and
- Ergodic graph theory, which is a very new and undeveloped subject, which roughly speaking seems to be concerned with the patterns within a measure-preserving action of the infinite permutation group , which is one of several models we have available to study infinite “limits” of graphs.
The two “discrete” (or “finitary”, or “quantitative”) fields of combinatorial number theory and graph theory happen to be related to each other, basically by using the Cayley graph construction; I will give an example of this shortly. The two “continuous” (or “infinitary”, or “qualitative”) fields of ergodic theory and ergodic graph theory are at present only related on the level of analogy and informal intuition, but hopefully some more systematic connections between them will appear soon.
On the other hand, we have some very rigorous connections between combinatorial number theory and ergodic theory, and also (more recently) between graph theory and ergodic graph theory, basically by the procedure of viewing the infinitary continuous setting as a limit of the finitary discrete setting. These two connections go by the names of the Furstenberg correspondence principle and the graph correspondence principle respectively. These principles allow one to tap the power of the infinitary world (for instance, the ability to take limits and perform completions or closures of objects) in order to establish results in the finitary world, or at least to take the intuition gained in the infinitary world and transfer it to a finitary setting. Conversely, the finitary world provides an excellent model setting to refine one’s understanding of infinitary objects, for instance by establishing quantitative analogues of “soft” results obtained in an infinitary manner. I will remark here that this best-of-both-worlds approach, borrowing from both the finitary and infinitary traditions of mathematics, was absolutely necessary for Ben Green and I in order to establish our result on long arithmetic progressions in the primes. In particular, the infinitary setting is excellent for being able to rigorously define and study concepts (such as structure or randomness) which are much “fuzzier” and harder to pin down exactly in the finitary world.
The question in extremal graph theory I wish to discuss here originates from Luca Trevisan; it shows that we still don’t know everything that we should about the “local” properties of large dense graphs.
Let be a large (undirected) graph, thus V is the vertex set with some large number n of vertices, and E is the collection of edges {x,y} connecting two vertices in the graph. We can allow the graph to have loops {x,x} if one wishes; it’s not terribly important for this question (since the number of loops is so small compared to the total number of edges), so let’s say there are no loops. We define three quantities of the graph G:
- The edge density , defined as the number of edges in G, divided by the total number of possible edges, i.e. ;
- The triangle density , defined as the number of triangles in G (i.e. unordered triplets {x,y,z} such that {x,y},{y,z}, {z,x} all lie in G), divided by the total number of possible triangles, namely ;
- The diamond density , defined as the number of diamonds in G (i.e. unordered pairs { {x,y,z}, {x,y,w} } of triangles in G which share a common edge), divided by the total number of possible diamonds, namely .
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