Related papers: Multiple ergodic averages in abelian groups and Kh…

Examining multiple ergodic averages whose iterates are integer parts of real valued polynomials for totally ergodic systems, we provide various characterizations of total joint ergodicity, meaning that an average converges to the "expected"…

Let $G$ be a nonabelian finite group and let $d$ be an irreducible character degree of $G$. Then there is a positive integer $e$ so that $|G| = d(d+e)$. Snyder has shown that if $e > 1$, then $|G|$ is bounded by a function of $e$. This…

Let $M$ be a semifinite von Neumann algebra and $T$ a positive contraction on both $L^1(M)$ and $L^\infty(M)$. We consider ergodic averages along a random sparse subsequence determined by independent Bernoulli variables $(X_n)_{n\geq 1}$…

Let $\F$ a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm $U^k(\X)$ for an ergodic action $(T_g)_{g \in \F^\omega}$ of the infinite abelian group $\F^\omega$ on a probability space…

B. Host and B. Kra (2005) have introduced the characteristic factors for studying cubic ergodic averages. These factors allow one to resolve, in particular, multiple recurrence problems introduced by H. Furstenberg (1977). We show here that…

Let (G,+) be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism t_g is ergodic. The main result of this paper is a new ergodic theorem for functions in L^1(G,M),…

In the first part of the paper the natural scheme for proving noncommutative individual ergodic theorems for multiple sequences is described and applied to obtain results on unrestricted convergence of multiaverages. In the second part…

Walsh has recently proved the norm convergence of all nonconventional ergodic averages involving polynomial sequences in discrete nilpotent acting groups. He deduces this convergence from an equivalent, `finitary' assertion of stability…

Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation…

We consider random fields indexed by finite subsets of an amenable discrete group, taking values in the Banach-space of bounded right-continuous functions. The field is assumed to be equivariant, local, coordinate-wise monotone, and almost…

A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result…

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges in $L^{2}$ to the…

We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every…

In this paper we combine methods from additive combinatorics and Diophantine geometry to study the generalised sum-product phenomenon in algebraic groups. As an application of this circle of ideas, we resolve a conjecture of Bremner on…

This paper is devoted to the study of noncommutative ergodic theorems for connected amenable locally compact groups. For a dynamical system $(\mathcal{M},\tau,G,\sigma)$, where $(\mathcal{M},\tau)$ is a von Neumann algebra with a normal…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0 < a < 1/2$, and let $p(n) = n^{1+\epsilon}$, $0 < \epsilon < 1$. We prove that, almost surely, for every…

We collect problems on recurrence for measure preserving and topological actions of a countable abelian group, considering combinatorial versions of these problems as well. We solve one of these problems by constructing, in…

Let $\mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $\mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+\mathcal{A}(q)$. On the other hand, let $\Bbbk$ denote the algebraic closure of $\mathbb{F}_{q}$, and…

We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite…

It is shown that there exist a subsequence for which the multiple ergodic averages of commuting invertible measure preserving transformations of a Lebesgue probability space converge almost everywhere provided that the maps are weakly…