Vitaly Bergelson

Polynomial extensions of van der Waerden’s and Szemerédi’s theorems

An extension of the classical van der Waerden and Szemeredi the- orems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: Let S ⊆ Zl be a set of positive upper Banach density, let p1(n), . . . , pk(n) be polynomials with rational coefficients taking integer values on the integers and satisfying pi(0) = 0, i = 1, . . . , k; then for any v1, . . . , vk ∈ Zl there exist an integer n and a vector u ∈ Zl such that u+ pi(n)vi ∈ S for each i ≤ k. Department of Mathematics, Ohio State University, Columbus, Ohio 43210 E-mail address: vitaly@math.ohio-state.edu Department of Mathematics, Technion, Haifa 23000, Israel E-mail address: sashal@techunix.technion.ac.il Current address: Department of Mathematics, Stanford University, Stanford, California 94305 E-mail address: leibman@math.stanford.edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Weakly mixing PET

Suppose that ( X , ℬ, μ) is a probability measure space and T is an invertible measure perserving transformation of ( X , ℬ, μ). T is called weakly mixing if for any two sets A 1 A 2 ∈ ℬ one has:

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of \({{\mathbb {F}^{\infty}_{p}}}\)

Let

IP-sets and polynomial recurrence

{\mathbb {F}}

Chapter 12 – Combinatorial and Diophantine Applications of Ergodic Theory

a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm Uk(X) for an ergodic action

On IP* sets and central sets

{(T_{g})_{{g} \in \mathbb {F}^{\omega}}}