Iterated differences sets, diophantine approximations and applications

TL;DR

This paper explores the properties of sets defined by polynomial differences and their applications in ergodic theory and combinatorics, providing new characterizations and variants of classical theorems.

Contribution

It introduces novel results on iterated difference sets for odd polynomials and applies these to characterize weakly mixing systems and extend Furstenberg-Sárközy theorem.

Findings

New characterization of weakly mixing systems

A new variant of Furstenberg-Sárközy theorem

Applications to ergodic theory and combinatorics

Abstract

Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal R(v,\epsilon)=\{n\in\mathbb{N}\,|\,\|v(n)\|{<\epsilon\}}$ where $\|\cdot\|$ denotes the distance to the closest integer. We then apply the new diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-S\'ark\"ozy theorem.