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Tim Austin

Publications and source records attributed to Tim Austin.

52 records · Page 3Linked to original sources
arXiv2012

On discontinuities of cocycles in cohomology theories for topological groups

This paper studies Moore's measurable cohomology theory for locally compact groups and Polish modules. An elementary dimension-shifting argument is used to show that all classes in that theory have representatives with considerable extra topological structure beyond measurability. Using this, for certain target modules one can also construct a direct comparison map with a different cohomology theory for topological groups defined by Segal, and show that this map is an isomorphism.

math.GR↗
arXiv2011

Deducing the Density Hales-Jewett Theorem from an infinitary removal lemma

We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: For any $δ> 0$ there is some $N_0 \geq 1$ such that whenever $A \subseteq [k]^N$ with $N \geq N_0$ and $|A|\geq δk^N$, $A$ contains a \textbf{combinatorial line}: that is, for some $I \subseteq [N]$ nonempty and $w_0 \in [k]^{[N]\setminus I}$ we have A \supseteq \{w: w|_{[N]\setminus I} = w_0, w|_I = \rm{const.}\}. Following Furstenberg and Katznelson, we first show that this result is equivalent to a `multiple recurrence' assertion for a class of probability measures enjoying a certain kind of stationarity. However, we then give a quite different proof of this latter assertion through a reduction to an infinitary removal lemma in the spirit of Tao's work on infinite random hypergraphs (and also its recent re-interpretation in a new proof of the multidimensional Szemeredi Theorem by the present author). This reduction is based on a structural analysis of these stationary laws closely analogous to the classical representation theorems for various partial exchangeable stochastic processes in the sense of Hoover, Aldous and Kallenberg. However, the underlying combinatorial arguments used to prove this theorem are rather different from those required to work with exchangeable arrays, and involve crucially an observation that arose during ongoing work by a collaborative team of authors to give a purely finitary proof of the above theorem.

math.PR↗
arXiv2010

Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Let $\H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function $f:\H\to X$ there exist $x,y\in \H$ with $d_W(x,y)$ arbitrarily large and \begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim \left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation} We also show that any embedding into $X$ of a ball of radius $R\ge 4$ in $\H$ incurs bi-Lipschitz distortion that grows at least as a constant multiple of \begin{equation}\label{eq:dist abs} \left(\frac{\log R}{\log\log R}\right)^{1/p}. \end{equation} Both~\eqref{eq:comp abs} and~\eqref{eq:dist abs} are sharp up to the iterated logarithm terms. When $X$ is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to~\eqref{eq:comp abs} and~\eqref{eq:dist abs} which are sharp up to a universal constant.

math.MG↗
arXiv2010

Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems

The Furstenberg recurrence theorem (or equivalently, Szemerédi's theorem) can be formulated in the language of von Neumann algebras as follows: given an integer $k \geq 2$, an abelian finite von Neumann algebra $(\M,τ)$ with an automorphism $α: \M \to \M$, and a non-negative $a \in \M$ with $τ(a)>0$, one has $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \Re τ(a α^n (a) ... α^{(k-1)n} (a)) > 0$; a subsequent result of Host and Kra shows that this limit exists. In particular, $\Re τ(a α^n (a) >... α^{(k-1)n} (a)) > 0$ for all $n$ in a set of positive density. From the von Neumann algebra perspective, it is thus natural to ask to what extent these results remain true when the abelian hypothesis is dropped. All three claims hold for $k = 2$, and we show in this paper that all three claims hold for all $k$ when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for $k=3$ when the von Neumann algebra is ergodic. However, we show that the first claim can fail for $k=3$ even with ergodicity, the second claim can fail for $k \geq 4$ even assuming ergodicity, and the third claim can fail for $k=3$ without ergodicity, or $k \geq 5$ and odd assuming ergodicity. The second claim remains open for non-ergodic systems with $k=3$, and the third claim remains open for ergodic systems with $k=4$.

math.OA↗
arXiv2010

Multiple recurrence and the structure of probability-preserving systems

In 1975 Szemerédi proved the long-standing conjecture of Erdős and Turán that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemerédi's proof was entirely combinatorial, but two years later Furstenberg gave a quite different proof of Szemerédi's Theorem by first showing its equivalence to an ergodic-theoretic assertion of multiple recurrence, and then bringing new machinery in ergodic theory to bear on proving that. His ergodic-theoretic approach subsequently yielded several other results in extremal combinatorics, as well as revealing a range of new phenomena according to which the structures of probability-preserving systems can be described and classified. In this work I survey some recent advances in understanding these ergodic-theoretic structures. It contains proofs of the norm convergence of the `nonconventional' ergodic averages that underly Furstenberg's approach to variants of Szemerédi's Theorem, and of two of the recurrence theorems of Furstenberg and Katznelson: the Multidimensional Multiple Recurrence Theorem, which implies a multidimensional generalization of Szemerédi's Theorem; and a density version of the Hales-Jewett Theorem of Ramsey Theory.

math.DS↗
arXiv2010

An alternative ending to "Pleasant extensions retaining algebraic structure''

The culmination of the papers (arXiv:0905.0518, arXiv:0910.0909) was a proof of the norm convergence in $L^2(μ)$ of the quadratic nonconventional ergodic averages \frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)\quad\quad f_1,f_2\in L^\infty(μ) associated to an arbitrary probability-preserving \bbZ^2-system (X,μ,T_1,T_2). This is a special case of the Bergelson-Leibman conjecture on the norm convergence of polynomial nonconventional ergodic averages. That proof relied on some new machinery for extending probability-preserving $\bbZ^d$-systems to obtain simplified asymptotic behaviour for various nonconventional averages such as the above. The engine of this machinery is formed by some detailed structure theorems for the `characteristic factors' that are available for some such averages after ascending to a suitably-extended system. However, these new structure theorems underwent two distinct phases of development, separated by the discovery of some new technical results in Moore's cohomology theory for locally compact groups (arXiv:1004.4937). That discovery enabled a significant improvement to the main structure theorem (Theorem 1.1 in (arXiv:0905.0518)), which in turn afforded a much shortened proof of convergence. However, since the proof of convergence using the original structure theorem required some quite different ideas that are now absent from these other papers, I have recorded it here in case it has some independent interest.

math.DS↗
arXiv2009

Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications

We study a generalized notion of a homogeneous skew-product extension of a probability-preserving system in which the homogeneous space fibres are allowed to vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of `direct integral' for a `measurable family' of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions, including non-ergodic versions of the results of Mackey describing ergodic components of such extensions, of the Furstenberg-Zimmer Structure Theory and of results of Mentzen describing the structure of automorphisms of such extensions when they are relatively ergodic. We then offer applications to two structural results for actions of several commuting transformations: firstly to describing the possible joint distributions of three isotropy factors corresponding to three commuting transformations; and secondly to describing the characteristic factors for a system of double nonconventional ergodic averages. Although both applications are modest in themselves, we hope that they point towards a broader usefulness of this formalism in ergodic theory.

math.DS↗
arXiv2009

Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles

We show that under some natural ergodicity assumptions extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L^\infty-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank modules associated to an extension of measure-preserving systems in the setting of a non-singular base.

math.DS↗
arXiv2009

On the testability and repair of hereditary hypergraph properties

Recent works of Alon-Shapira and Rödl-Schacht have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally". In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs. In the case of undirected graphs, we extend the result to continuous graphs on probability spaces, and show that the repair algorithm is "local" in the sense that it only depends on a bounded amount of data; in particular, the graph can be repaired in a time linear in the number of edges. We also show that local repairability also holds for monotone or partite hypergraph properties (this latter result is also implicitly in work of Ishigami). In the negative direction, we show that local repairability breaks down for directed graphs, or for undirected 3-uniform hypergraphs. The reason for this contrast in behavior stems from (the limitations of) Ramsey theory.

math.CO↗
arXiv2009

Deducing the multidimensional Szemeredi Theorem from an infinitary removal lemma

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T_1, T_2, >..., T_d: \bbZ\curvearrowright (X,§,μ), and so, via the Furstenberg correspondence principle introduced in, a new proof of the multi-dimensional Szemeredi Theorem. We bypass the careful manipulation of certain towers of factors of a probability-preserving system that underlies the Furstenberg-Katznelson analysis, instead modifying an approach recently developed for the study of convergence of nonconventional ergodic averages to pass to a large extension of our original system in which this analysis greatly simplifies. The proof is then completed using an adaptation of arguments developed by Tao for his study of an infinitary analog of the hypergraph removal lemma. In a sense, this addresses the difficulty, highlighted by Tao, of establishing a direct connection between his infinitary, probabilistic approach to the hypergraph removal lemma and the infinitary, ergodic-theoretic approach to Szemeredi's Theorem set in motion by Furstenberg.

math.DS↗
arXiv2009

On the norm convergence of nonconventional ergodic averages

We offer a generalization of the recent result of Tao (building on earlier results of Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler) that the nonconventional ergodic averages associated to an arbitrary number of commuting probability-preserving transformations always converge to some limit in L^2. We prove the corresponding result for a collection of commuting actions of a larger discrete Abelian group, and gives convergence that is uniform in the start-point of the averages. While Tao's proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.

math.DS↗
arXiv2008

On the geometry of a class of invariant measures and a problem of Aldous

In his 1985 survey of notions of exchangeability, Aldous introduced a form of exchangeability corresponding to the symmetries of the infinite discrete cube, and asked whether these exchangeable probability measures enjoy a representation theorem similar to those for exchangeable sequences, arrays and set-indexed families. In this note we to prove that, whereas the known representation theorems for different classes of partially exchangeable probability measure imply that the compact convex set of such measures is a Bauer simplex (that is, its subset of extreme points is closed), in the case of cube-exchangeability it is a copy of the Poulsen simplex (in which the extreme points are dense). This follows from the arguments used by Glasner and Weiss' for their characterization of property (T) in terms of the geometry of the simplex of invariant measures for associated generalized Bernoulli actions. The emergence of this Poulsen simplex suggests that, if a representation theorem for these processes is available at all, it must take a very different form from the case of set-indexed exchangeable families.

math.PR↗
arXiv2008

On exchangeable random variables and the statistics of large graphs and hypergraphs

De Finetti's classical result of [18] identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous [1,2,3], Hoover [41,42], Kallenberg [44] and Kingman [47]. On the other hand, such exchangeable laws were first related to questions from combinatorics in an independent analysis by Fremlin and Talagrand [29], and again more recently in Tao [62], where they appear as a natural proxy for the `leading order statistics' of colourings of large graphs or hypergraphs. Moreover, this relation appears implicitly in the study of various more bespoke formalisms for handling `limit objects' of sequences of dense graphs or hypergraphs in a number of recent works, including Lovász and Szegedy [52], Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi [17], Elek and Szegedy [24] and Razborov [54,55]. However, the connection between these works and the earlier probabilistic structural results seems to have gone largely unappreciated. In this survey we recall the basic results of the theory of exchangeable laws, and then explain the probabilistic versions of various interesting questions from graph and hypergraph theory that their connection motivates (particularly extremal questions on the testability of properties for graphs and hypergraphs). We also locate the notions of exchangeability of interest to us in the context of other classes of probability measures subject to various symmetries, in particular contrasting the methods employed to analyze exchangeable laws with related structural results in ergodic theory, particular the Furstenberg-Zimmer structure theorem for probability-preserving $\mathbb {Z}$-systems, which underpins Furstenberg's ergodic-theoretic proof of Szemerédi's Theorem. The forthcoming paper [10]--hereditarytest will make a much more elaborate appeal to the link between exchangeable laws and dense (directed) hypergraphs to establish various results in property testing.

math.PR↗
arXiv2007

The wreath product of Z with Z has Hilbert compression exponent 2/3

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all $α\ge 0$ such that there exists a Lipschitz mapping $f:G\to L_2$ and a constant $c>0$ such that for all $x,y\in G$ we have $\|f(x)-f(y)\|_2\ge cd(x,y)^α.$ In \cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\Z\bwr \Z $ is at most $\frac34$, and in \cite{NP07} was proved that this exponent is at least $\frac23$. Here we show that $\frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type.

math.MG↗
arXiv2007

The Euclidean distortion of the lamplighter group

We show that the cyclic lamplighter group $C_2 \bwr C_n$ embeds into Hilbert space with distortion ${\rm O}(\sqrt{\log n})$. This matches the lower bound proved by Lee, Naor and Peres in \cite{LeeNaoPer}, answering a question posed in that paper. Thus the Euclidean distortion of $C_2 \bwr C_n$ is $Θ(\sqrt{\log n})$. Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni, Maurey and Mityagin \cite{AhaMauMit} and by Gromov (see \cite{deCTesVal}), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.

math.MG↗