Multiple ergodic theorems for arithmetic sets

TL;DR

This paper extends multiple recurrence and convergence theorems in ergodic theory to subsets of integers with specific arithmetic properties, using multiplicative functions and structural results to analyze ergodic averages.

Contribution

It introduces new arithmetic restrictions into multiple ergodic theorems and employs recent structural results on multiplicative functions for analysis.

Findings

Generalization of polynomial Szemeredi theorem with arithmetic restrictions

Analysis of multiple ergodic averages with multiplicative weights

Establishment of convergence results for these weighted averages

Abstract

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter to those integers that have an even number of distinct prime factors, or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analysed using a recent structural result for bounded multiplicative functions proved by the authors.