Sets of transfer times with small densities

TL;DR

This paper characterizes the structure of transfer time sets with small densities in ergodic systems, showing they are either periodic or Sturmian, extending classical sumset theorems to an ergodic context.

Contribution

It introduces an ergodic-theoretic extension of sumset theorems, classifying transfer time sets with small densities as either periodic or Sturmian.

Findings

Transfer time sets with small densities are either periodic or Sturmian.

The results extend classical sumset theorems to ergodic systems.

The proofs utilize a correspondence principle for action sets.

Abstract

We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Our proofs are based on a correspondence principle for action sets which was developed previously by the first two authors.