Multiple ergodic averages for variable polynomials

TL;DR

This paper proves convergence of multiple ergodic averages for variable polynomials under certain conditions, extending classical results and addressing an open problem in ergodic theory and number theory.

Contribution

It establishes convergence of variable polynomial ergodic averages with an additional assumption, advancing understanding of multiple ergodic averages and their applications.

Findings

Convergence of averages along integers for good variable polynomials.

Extension of classical recurrence and combinatorial results.

Analysis of averages along prime numbers.

Abstract

In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted by Frantzikinakis (Problem 10, "Some open problems on multiple ergodic averages. Bulletin of the Hellenic Mathematical Society. 60 (2016), 41-90"). Corresponding averages along prime numbers are studied too. These general convergence results imply various variable extensions of classical recurrence, combinatorial and number theoretical results which are presented as well.