Abstracts
Vitaly Bergelson
Pointwise joint ergodicity of number theoretic maps.
ABSTRACT: We will present some of the recent joint work with Younghwan Son on norm and pointwise convergence of multiple ergodic averages which involve jointly ergodic transformations. In particular, we will describe an ergodic approach to the study of joint normality of representations of numbers. For example, for any integer b ≥ 2, almost every number x ∈ [0, 1) is jointly normal with respect to the b-expansion and continued fraction expansion. This fact is a corollary of a general result which deals with pointwise joint ergodicity which takes place for a wide variety of number theoretic maps. We will also discuss some natural conjectures pertaining to pointwise convergence of multiple ergodic averages.Thomas Bloom
New approaches to three-term arithmetic progressions
ABSTRACT: The study of sets without three-term arithmetic progressions is central to additive combinatorics. In February this year Kelley and Meka sent a shockwave through the community with a new upper bound for the size of sets without three-term arithmetic progressions, far better than any previously available. Moreover, their approach is almost entirely 'physical', in contrast to the Fourier analytic methods that have previously been the most useful. I will discuss these new ideas, comparing to the previous Fourier approach, and discuss a small improvement (in joint work with Olof Sisask).Christina Giannitsi
Why should we care about averaging operators?
ABSTRACT: We have studied various averaging operators of discrete functions, inspired by number theory, and showed that they satisfy ℓp improving and maximal bounds. The maximal bounds are obtained via sparse domination results for p∈(1,2), which imply boundedness on ℓp(w) for p∈(1,∞), for all weights w in the Muckenhoupt Ap class. The purpose of the talk is to motivate our work and put it in the context of our long term research goals. We start by looking at averages along the integers weighted by the divisor function d(n), and obtain a uniform, scale free ℓp-improving estimate for p∈(1,2). We also show that the associated maximal function satisfies (p,p) sparse bounds for p∈(1,2). We move on to study averages along primes in arithmetic progressions, and establish improving and maximal inequalities for these averages, that are uniform in the choice of progression. The uniformity over progressions imposes several novel elements on our approach. Lastly, we generalize our setting in the context of number fields, by considering averages over the Gaussian primes. Finally, we explore the connections of our work to number theory: Fix an interval ω⊂T. There is an integer Nω, so that every odd integer n with norm N(n)>Nω is a sum of three Gaussian primes with arguments in ω. This is the weak Goldbach conjecture. A density version of the strong Goldbach conjecture is proved, as well.
This is based on past and current work with Ben Krause, Michael Lacey, Hamed Mousavi and Yaghoub Rahimi.Marina Iliopoulou
Some remarks on the Mizohata-Takeuchi conjecture
ABSTRACT: The extension operator E of harmonic analysis is the Fourier transform of functions defined on curved hypersurfaces in R^n. The restriction conjecture claims that, when these hypersurfaces have non-vanishing Gaussian curvature, E is L^{2n/(n-1)}-bounded (which would imply that the level sets of the extension operator are small). On the other hand, the Mizohata-Takeuchi conjecture aims to understand the shape of these level sets, by establishing L^2-weighted bounds for E. An interesting intermediate result would be to establish an analogue of the Mizohata-Takeuchi conjecture, but for L^{2n/(n-1)}-bounds instead. In this talk, we will discuss this "mix" of the two conjectures, and will provide a proof when n=2.
This is work in progress, joint with A. Carbery and B. Shayya.Alex Ionescu
Polynomial averages and pointwise ergodic theorems on nilpotent groups
ABSTRACT: I will talk about some recent work, in collaboration with Akos Magyar, Mariusz Mirek, and Tomasz Szarek, on pointwise convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. We also establish corresponding maximal inequalities on $L^p$ for $p\in (1,\infty]$ and $\rho$-variational inequalities on $L^2$ for $\rho\in(2,\infty)$. This gives an affirmative answer to the Furstenberg–Bergelson–Leibman conjecture for polynomial ergodic averages in discrete nilpotent groups of step two.Bryna Kra
Infinite patterns in large sets of integers
ABSTRACT: Resolving a conjecture of Erdos and Turan from the 1930's, in the 1970's Szemeredi showed that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg used Ergodic Theory to gave a new proof of this result, leading to the development of combinatorial ergodic theory. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. We discuss recent developments for infinite patterns leading to the resolution of conjectures of Erdos.
This is joint work with Joel Moreira, Florian Richter, and Donald Robertson.Borys Kuca
Multiple ergodic averages along polynomials for systems of commuting transformations
ABSTRACT: Furstenberg’s dynamical proof of the Szemerédi theorem initiated a thorough examination of multiple recurrence phenomena; in doing so, it laid the grounds for combinatorial ergodic theory. Multiple recurrence results are usually deduced from structural characterisations of multiple ergodic averages - analytic objects that generalise the classical Birkhoff averages. Of special interest are averages of commuting transformations with polynomial iterates, which play a central role in the polynomial Szemerédi theorem of Bergelson and Leibman. Their norm convergence has been established in a celebrated paper of Walsh, but for a long time, little more has been known about the form of the limit. A recent outburst of activity on this topic led to new structural results on the limits of such averages, bringing resolution of several previously intractable problems. I will present some of these recent developments with the emphasis on joint works with Nikos Frantzikinakis.Joel Moreira
Multidimensional extensions of a theorem of Weyl on uniform distribution
ABSTRACT: The theory of uniform distribution modulo 1 started with the work of Weyl, who proved several fundamental results. One of them states that given an increasing sequence (a_n) of integers, for almost every real number x, the sequence (a_nx) is uniformly distributed modulo 1. Applying this result with the sequence a_n=2^n one obtains a version of the pointwise ergodic theorem for the doubling map. There are several possible ways to extend Weyl's thrown, and we will explore some directions regarding multidimensional extensions.
This talk is based on ongoing joint work with Vitaly Bergelson.Amos Nevo
Intrinsic Diophantine approximation on homogeneous spaces
ABSTRACT: Let G be a Lie group, L a lattice in G, and H a closed subgroup of G. Suppose that L acts on the homogeneous space G/H with dense orbits. Naturally, we would like to measure how dense these orbits actually are. Departing from traditional classical Diophantine approximation, we will focus on the case where G is a non-amenable group, for example the general linear group or the general affine group. We will present a solution to this problem for lattice actions on a large class of homogeneous spaces, including a sufficient condition for when an optimal result holds, and give some examples. We will then briefly describe the truly extensive scope of this set-up, and explain some more refined problems related to gauges of denseness, equidistribution and discrepancy, as time permits. Three basic motivating examples to keep in mind that will be mentioned are : 1) the linear action of SL(2,Z) on the punctured plane, 2) the affine action of SL(2,Z)xZ^2 on the plane. 3) The action of SL(2,Q) on SL(2,R) by translations.
Based on joint work with Alex Gorodnik, Anish Ghosh and Mikolaj Fraczyk.Sean Prendiville
A multidimensional version of a nonlinear Roth theorem
ABSTRACT: A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form x, x+y, x+y^2. We discuss a multidimensional version of this result, with a focus on quantitative issues.Donald Robertson
Disintegrating product measures for sumsets
ABSTRACT: In this talk we describe how ergodic decompositions of product measures can be refined, and how these refinements can be used to prove the existence of sumsets in sets of positive density.
Based on joint work with Bryna Kra, Joel Moreira and Florian Richter.Florian Richter
Ergodic theorems for multiplicative actions of the integers
ABSTRACT: One of the fundamental challenges in number theory is to understand the multiplicative structure of the integers. In this talk we explore a dynamical approach to this topic. By studying ergodic averages of multiplicative actions of the integers, we arrive at a new ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to various other results and inquiries concerning the ergodic behavior of multiplicative actions, including an extended form of Sarnak's Mobius disjointness conjecture and questions regarding the spectral structure of such systems with connections to Fourier analysis.Tomasz Szarek
Pointwise ergodic theorems and Waring type problem on nilpotent groups
ABSTRACT: The famous Furstenberg-Bergelson-Leibman conjecture says that the ergodic averages converge pointwise almost everywhere provided that the underlying measure preserving transformations generate a nilpotent group. The purpose of this talk is to discuss recent progress towards solving this conjecture.
The talk will be based on joint works with Alexandru Ionescu, Akos Magyar and Mariusz Mirek.Máté Wierdl
What can be the limit of ergodic averages?
ABSTRACT: Let S := (s_1 < s_2 < . . .) be a strictly increasing sequence of positive integers so that in every dynamical system (X,m,T) and f ∈ L^2 (X), the sequence (1/N)∑n≤N f(T sn x) N∈N of ergodic averages along (sn) converge in L^2-norm. What can be the limit? We know that in case s_n = n and an ergodic T the limit is \int_X f dm. In case s_n = n^2 or when s_n is the nth prime number, the limit is again \int_X f dm provided T is totally ergodic, that is, T^k is ergodic for every positive integer k. By the spectral theorem, the limit being the integral in totally ergodic systems is equivalent with \lim_N (1/N) ∑n≤N e(snα) = 0 for every irrational α, where we used Weyl’s notation e(β) := e^{2πiβ} . This limit being 0 is equivalent with saying that the limit Borel probability measure µ_{S,α}:=\lim_N(1/N)∑n≤Nδ_{s_nα} for irrational α is the Haar-Lebesgue measure on the torus T:=[0,1). Can this limit measure be other Borel probability measures?
It turns out that μ_{S,α} must be a continuous measure if α is irrational. Our main question hence is: Can the limit measure μ_{S,α} be any continuous measure for irrational α? Well, if the answer was yes, then it would confirm Furstenberg’s ×2 × 3 conjecture. But no, we do not know the answer to this question, though we have been trying for a while now. We have figured out quite a few things, though. We know, for example, that the answer will depend on the irrational α, and along the way we obtain a new characterization of Rajchman measures, that is, measures with vanishing Fourier coefficients at infinity. In the talk, we will necessarily see many new good sequences, and for this conference the main focus would be to see which of these sequences are good pointwise. The solution of these problems would likely require adjustments of the circle method and we’ll explain why and how.Jim Wright
Oscillation inequalities in ergodic theory and analysis.
ABSTRACT: We revisit the topic of oscillation inequalities in almost everywhere pointwise convergence problems for a couple of reasons. First, we develop a theoretical framework for oscillations to put these inequalities in some context and furthermore, we claim that the oscillation semi-norm is the only viable tool to efficiently study multiparameter pointwise convergence problems with arithmetic features.
This is joint work with M. Mirek and T. Szarek.