Related papers: Ergodic averages of commuting transformations with…

We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…

Let $\{T^z\}$ be an ergodic action of the group $Z^n$ by automorphisms of the probability space $(X,m)$, $\sum_{i}^\infty a_i<\infty$, $a_i>0$. For any sequence $M_k\to +\infty$ there exist $N_k>M_k$ and a function $ f\in L_1(X,m)$ such…

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,\S,\mu), and so, via the Furstenberg correspondence…

Given an ergodic dynamical system $(X, \mathcal{B}, \mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, \mu)$, the ergodic averages \[ \frac{1}{\pi(N)} \sum_{p \in \mathbb{P}_N} f\big(T^p…

Let $\mathcal{P}$ be an (unbounded) countable multiset of primes, let $G=\bigoplus_{p\in P}\mathbb{F}_p$. We study the $k$'th universal characteristic factors of an ergodic probability system $(X,\mathcal{B},\mu)$ with respect to some…

We study multiple ergodic averages along IP sets, meaning we restrict iterates in the averages to all finite sums of some infinite sequence of natural numbers. We give criteria for convergence and divergence in mean of these multiple…

Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove…

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our…

We calculate a certain mean-value of meromorphic functions by using specific ergodic transformations, which we call affine Boolean transformations. We use Birkhoff's ergodic theorem to transform the mean-value into a computable integral…

We generalize a result of Matom\"aki, Radziwi{\l}{\l}, and Tao, by proving an averaged version of a conjecture of Chowla and a conjecture of Elliott regarding correlations of the Liouville function, or more general bounded multiplicative…

We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(\lambda_k^{(1)}t_1+...+\lambda_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{\lambda_n^{(i)}:1\leq i\leq s\}$ are sequences of real…

In this paper, we extend Bourgain's double recurrence result to the Wiener-Wintner averages. Let $(X, \mathcal{F}, \mu, T)$ be a standard ergodic system. We will show that for any $f_1, f_2 \in L^\infty(X)$, the double recurrence…

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…

We introduce an ergodic approach to the study of {\em joint normality} of representations of numbers. For example, we show that for any integer $b \geq 2$ almost every number $x \in [0,1)$ is jointly normal with respect to the $b$-expansion…

We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the…

In this article, we continue the structural study of factor maps betweeen symbolic dynamical systems and the relative thermodynamic formalism. Here, one is studying a factor map from a shift of finite type $X$ (equipped with a potential…

Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under…

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \}…

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.…

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector…