Related papers: Ergodic averages for sparse sequences along primes

We examine multiple ergodic averages of commuting transformations with polynomial iterates in which the polynomials may be pairwise dependent. In particular, we show that such averages are controlled by the Gowers-Host-Kra seminorms…

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good…

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to…

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…

Let $T$ be a power bounded Hilbert space operator without unimodular eigenvalues. We show that the subsequential ergodic averages $N^{-1}\sum_{n=1}^N T^{a_n}$ converge in the strong operator topology for a wide range of sequences $(a_n)$,…

This article is devoted to studying individual ergodic theorems for subsequential weighted ergodic averages on the noncommutative Lp-spaces associated to a semifinite von Neumann algebra M. In particular, we establish the convergence of…

We study convergence of ergodic averages along squares with polynomial weights. For a given polynomial $P\in \mathbb{Z}[\cdot]$, consider the set of all $\theta\in[0,1)$ such that for every aperiodic system $(X,\mu, T)$ there is a function…

We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a…

In the framework of statistical mechanics the properties of macroscopic systems are deduced starting from the laws of their microscopic dynamics. One of the key assumptions in this procedure is the ergodic property, namely the equivalence…

We show that once $\theta>17/30$, every sufficiently long interval $[x,x+x^\theta]$ contains many $k$-term arithmetic progressions of primes, uniformly in the starting point $x$. More precisely, for each fixed $k\ge3$ and $\theta>17/30$,…

We prove that if $\mu_n$ are probability measures on $Z$ such that $\hat \mu_n$ converges to 0 uniformly on every compact subset of $(0,1)$, then there exists a subsequence $\{n_k\}$ such that the weighted ergodic averages corresponding to…

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic…

We consider ergodic multiflows on a probability space. The general theorem on universal averaging for multiflows is applied to averaging along manifolds in $R^n$.

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\Sigma,\mu)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field…

For a jointly measurable probability-preserving action $\tau:\mathbb{R}^D\curvearrowright (X,\mu)$ and a tuple of polynomial maps $p_i:\mathbb{R}\to \mathbb{R}^D$, $i=1,2,...,k$, the multiple ergodic averages \[ \frac{1}{T}\int_0^T…

The arithmetic average of the first $n$ primes, $\bar p_n = {1\over n} \sum_{i=1}^n p_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n \to \bar p_n$ is extremely easy to invert, $p_n = n\bar p_n -…

In this paper we combine the general tools developed in (arXiv:0905.0518) with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss, Host and Kra and Ziegler to study the…

In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1, 3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local…

We examine the limiting behavior of multiple ergodic averages associated with arithmetic progressions whose differences are elements of a fixed integer sequence. For each $\ell$, we give necessary and sufficient conditions under which…