Related papers: Some open problems on multiple ergodic averages
In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted…
We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…
Using an ergodic inverse theorem obtained in our previous paper, we obtain limit formulae for multiple ergodic averages associated with the action of $\mathbb{F}_{p}^{\omega}$. From this we deduce multiple Khintchine-type recurrence results…
We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain…
The survey presents the main developments obtained over the last decade regarding pointwise ergodic theorems for measure preserving actions of locally compact groups. The survey includes an exposition of the solutions to a number of long…
These notes are based on a course for a general audience given at the Centro de Modeliamento Matem\'atico of the University of Chile, in December 2004. We study the mean convergence of multiple ergodic averages, that is, averages of a…
This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.
We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…
In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.
This note presents a discussion of the algebraic and combinatorial aspects of the theory of pure O-sequences. Various instances where pure O-sequences appear are described. Several open problems that deserve further investigation are also…
We find limits of some multiple ergodic averages, generalizing a result of Bergelson to the setting of two commuting transformations and actions of amenable groups.
In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena…
In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of $L^p$-norms and leave its a.s. convergence as an open problem.…
The aim of this survey is to present some aspects of multifractal analysis around the recently developed subject of multiple ergodic averages. Related topics include dimensions of measures, oriented walks, Riesz products etc.
We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the "magic extension" established recently…
We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and…
For a differential equation with interaction, we investigate its ergodic properties. We apply the obtained results to study the limiting behavior of braid invariants associated with the flow of solutions.
The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We…
We give a short combinatorial proof of the classical pointwise ergodic theorem for probability measure preserving $\mathbb{Z}$-actions. Our approach reduces the theorem to a tiling problem: tightly tile each orbit by intervals with desired…
We study the almost sure convergence of bilateral ergodic averages for not necessarily integrable functions and relate it to the ones of the forward and backward averages, hence complementing results of Wo\'s and the second named author. In…