Ergodic Theory and Combinatorics Conference University of Agder, Kristiansand

June 8 - 12, 2015

Abstracts of the talks

Tim Austin, Courant Institute, USA

Multiple recurrence and Ajtai-Szemeredi Theorems for products of groups.

I will sketch two recent and somewhat analogous pieces of work: a generalization of the Furstenberg-Katznelson Theorem to commuting actions of an amenable group, and genera- lizations of the Ajtai-Szemeredi Theorem to the Cartesian square of a quasirandom group. The ﬁrst answers a question of Bergelson, and the second gives a quantitative version of some recent results of Bergelson, Robertson and Zorin-Kranich. The second work also pro- ves a new Ajtai-Szemeredi-type theorem, for which an ergodic theoretic analog is unclear.

Mathias Beiglb¨ock, University of Vienna, Austria

Cyclical monotonicity and the ergodic theorem

A fundamental concept in the ﬁeld of mass transport is the notion of cyclical monotonicity: it provides a combinatorial characterization of optimal transport plans which is key to understanding the geometry of mass transport. We explain how this link between op- timality and its combinatorial counterpart can be understood through the ergodic theorem.

1 Daniel Berend, Ben-Gurion University, Israel

Reconstruction of Domino Tilings (joint with Yoav Bar-Sinai)

n+2 Dominosa is a puzzle game. Before the game begins, the 2 domino bones (0, 0), (0, 1), ..., (n, n) are placed on an (n+1)×(n+2) grid (constituting a perfect tiling). The domino bones are then being hidden, leaving only the numbers revealed. The purpose of the game is to recover the original tiling of the domino bones. The game raises many questions, such as how various algorithms perform and how many solutions a random puzzle has. We will discuss some of these questions.

Tomasz Downarowicz, Wroclaw University of Technology, Poland

Large sets of integers and hierarchy of mixing properties of measure-preserving systems

Manfred Einsiedler, ETH Zurich, Switzerland

Equidistribution of integer points on spheres and their orthogonal lattices

Refining a problem posed by Linnik we study primitive integer points on large spheres and the shape of the lattice in the orthogonal complement. As we will discuss in dimensions 6 and above the joint equidistribution of the direction of the integer point and the shape of the lattice is a corollary of the p-adic version of the Mozes-Shah theorem which in this form is due to Gorodnik and Oh. In the three-dimensional space a partial result (under some congruence conditions) can be proven as a corollary of a joining classification result for diagonal actions that is joint work with E. Lindenstrauss. In dimensions 4 and 5 the full equidistribution result can be obtained by using effective methods for semisimple adelic group actions. This is joint work with M. Aka, U. Shapira respectively with R. Ruehr, and P. Wirth.

Tanja Eisner, University of Leipzig, Germany

Weighted ergodic theorems

We discuss some classes of good weights for the pointwise ergodic theorem.

2 Nikos Frantzikinakis, University of Crete, Greece

Multiple correlation sequences and nilsequences.

Multiple correlation sequences appear naturally when studying recurrence and convergence phenomena in ergodic theory and related problems in combinatorics and number theory. Several open problems rest upon a detailed understanding of their intrinsic structure. We shall see that under very general hypothesis any such correlation sequence is the sum of a nilsequence and an error term that is small in uniform density. This result vastly gene- ralizes works of Bergelson-Host-Kra and Leibman and relies on much simpler arguments than previously used. We use this decomposition to prove some rather surprising convergence criteria for multiple ergodic averages and indicate also some implications related to multiple recurrence (the latter is based on joint work with B. Host). We will also mention a related open problem that is of particular importance in this area.

Hillel Furstenberg, Hebrew University of Jerusalem, Israel

Ergodic Ramsey Theory for Non-amenable Groups

For amenable groups one can imitate Ergodic Ramsey Theory in the spirit of V. Bergelson’s work without too much ado. For non-amenable groups one has to develop ergodic theory for situations where no measure is invariant; rather the notion of a stationary measure takes its place. As an example we present a Szemeredi theorem for the free group.

Neil Hindman, Howard University Washington, USA

Vitaly, Central Sets, and me

Central subsets of N were invented by Furstenberg around 1980, deﬁned in terms of uni- formly recurrent points and proximality. For reasons known only to him (and maybe God), Vitaly suggested in the late 1980’s that these sets might be exactly the members of minimal idempotents in the Stone-Cech compactiﬁcation of N. He turned out to be right and that characterization applies to arbitrary semigroups. This has led to a quarter of a century of entertainment and fruitful research for me which I will discuss in this talk.

3 Bernard Host, Universit´eParis-Est Marne-la-Vall´ee,France

Partition regularity and multiplicative functions.

Partition regularity is a classical notion in combinatorics: A collection of subsets of the integer is partition regular if for every ﬁnite partition of the integers at least one atom contains an element of the collection. Vitaly Bergelson has important contribution in this topic. On the other hand, multiplicative functions are a standard topic in number theory. In this talk we explain how a question of partition regularity led us to an unusual problem involving multiplicative functions, and we plan to say a few words about the methods needed to solve it. This is a joint work with Nikos Frantzikinakis.

Bryna Kra, Northwestern University, USA

Variations on topological recurrence

We discuss relations between sets of recurrence for diﬀerent systems and ways to formulate ﬁnite versions of recurrence. This is based on joint work with Bernard Host and Alejandro Maass.

Imre Leader, Cambridge University, UK

Pairwise sums in the reals

We show that there is a finite colouring of the reals such that there is no uncountable set with all pairwise sums the same colour. We also show that there is a finite colouring of the reals such that no infinite set X has X+X (meaning the pairwise sums from X, allowing repetition) monochromatic. And we give positive results for ‘nice’ colourings. (Joint work with Neil Hindman and Dona Strauss.)

Mariusz Lemanczyk, Nicolaus Copernicus University in Torun, Poland

B-free systems and their invariant measures

S Assume B = {b1, b2, ...} is a subset of natural numbers. Then the set MB := k≥1 bkZ is called the set of multiples of B and its complement FB := Z \ MB is the set of so called

4 Z B-free numbers. The characteristic function η := 1FB can be viewed as a point in {0, 1} , Z hence the closure of its orbit under the shift S yields a subshift Xη ⊂ {0, 1} . For example, when B = {p2 : p is prime}, we obtain so called square-free system. I will present a solution of the problem of invariant measure for B-free systems (S,Xη) assuming that the set B is P an inﬁnite set of pairwise coprime natural numbers (each bk ≥ 2) and k≥1 1/bk < +∞. Moreover, intrinsic ergodicity of such B-free systems will be shown to hold. The talk is based on my joint work with J. Kulaga-Przymus and B. Weiss.

Randall McCutcheon, University of Memphis, USA tba

Michal Misiurewicz, Indiana University-Purdue University Indianapolis, USA

Counting preimages

For rational maps on the Riemann sphere one of the standard ways of drawing the Julia set is to follow a random trajectory backward. Here “random” means that each time we choose randomly one of the preimages of the current point. Measures equidistributed on longer and longer pieces of such trajectory converge to the measure of maximal entropy. We investigate what happens if we apply this procedure to the subshifts of ﬁnite type and piecewise monotone interval maps, under the assumption of transitivity. It turns out that the limit still exists (for interval maps only almost everywhere), but usually it is not the measure with maximal entropy. This is joint work with Ana Rodrigues.

Hitoshi Nakada, Keio University, Japan

How to ﬁnd the absolutely continuous invariant measure for continued fraction maps

In 1888, A. Hurwitz introduced the nearest type continued fraction map for complex numbers. It is easy to show that there exists an absolutely continuous invariant probability measure for this map. In this talk, I will explain how to ﬁnd the explicit form of the density function of this invariant measure. (Joint work with Hiromi Ei and Rie Natsui)

5 Joe Rosenblatt, Indiana University-Purdue University Indianapolis, USA

Directional Behavior of Actions by Commuting Maps

Directional behavior of actions by commuting maps will be described in terms of spectral and intrinsic behavior. The intrinsic behavior has both averaging and point realizations that are closely related to Diophantine approximation. In addition, the generic behavior of actions and connections with thin sets in harmonic analysis will be discussed.

Imre Ruzsa, Alfr´edR´enyi Institute of Mathematics, Hungary

Our (losing) fight with the squares We try, unsuccesfully, to find a decent estimate for the number of residues modulo m so that no difference is a square. (with MátéMatolcsi)

Dona Strauss, University of Hull, UK

Additive and multiplicative idempotents in βN

The properties of idempotents in the Stone-Cech compactiﬁcation of a semigroup, have played a fruitful role in combinatorics. In particular, the fact that there are minimal multiplicative idempotents in βN in the closure of the set of minimal additive idempotents of βN, has led to the discovery of important new theorems. I shall discuss some of these theorems, and give a simple proof that there are no additive idempotents in βN in the closure of the set of multiplicative idempotents of βN. This answered a question posed by Vitaly Bergelson.

Terence Tao, UCLA, USA

Concatenation of Gowers uniformity and anti-uniformity norms, and applications

A function of two variables which is a polynomial of degree d1 in the ﬁrst variable, and a polynomial of degree d2 in the second variable, is necessarily a polynomial of degree d1 +d2 in both variables. We generalise such ”concatenation theoremsto the setting of Gowers anti-uniform functions, characteristic factors, or nilsequences. By duality, this also implies that functions which are globally Gowers uniform are necessarily locally Gowers uniform in most directions. This has applications for counting problems involving polynomial patterns

6 such as n, n + r, n + r2, for instance one can now obtain asymptotics for the number of such patterns consisting solely of primes. This is joint work with Tamar Ziegler.

Tom Ward, Durham University, UK

Mixing then and now

About 25 years ago one of the things Vitaly Bergelson and I talked about was mixing properties of commuting transformations. I will give a brief overview of some of the questions that existed then and what they have led to since.

Benjamin Weiss, Hebrew University of Jerusalem, Israel

The isomorphism problem for smooth dynamical systems

By a smooth dynamical system we mean a diffeomorphism of a compact manifold preserving a volume element. A few years ago in joint work with Matt Foreman and Dan Rudolph we showed that the abstract isomorphism relation between ergodic transformations is a complete analytic set. To date, it is not known whether or not all ergodic finite entropy transformations have smooth models. I will describe a joint project with Matt Foreman in which we show that the the isomorphism relation when restricted to ergodic Lebesgue measure preserving diffeomorphisms of the torus is also a complete analytic set.

Tamar Ziegler, Hebrew University of Jerusalem, Israel

Narrow polynomial progressions in primes

Pavel Zorin-Kranich, University of Bonn, Germany

Variation norm estimates in pointwise ergodic theorems

Over the last two years there appeared variation norm versions of Bourgain’s pointwise ergodic theorems: along primes, polynomials, for bilinear averages, and return times averages (the latter so far only in a dyadic model). The common enabling idea in these results is a variation norm multi-frequency lemma of Nazarov, Oberlin, and Thiele. I will demonstrate its use on the example of averages along primes.