Decomposition of random walk measures on the one-dimensional torus

TL;DR

This paper presents a decomposition theorem for measures on the one-dimensional torus, splitting any measure into parts that either rapidly equidistribute under certain random walks or are concentrated near few points.

Contribution

It introduces a novel measure decomposition on the torus that distinguishes between rapidly equidistributing parts and localized parts, advancing understanding of random walk behavior.

Findings

Decomposition of measures into two distinct types.

Rapid equidistribution of one component under random walk.

Concentration of the other component near few points.

Abstract

The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset $S$ of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $\mu_1$ has the property that the random walk with initial distribution $\mu_1$ evolved by the action of $S$ equidistributes very fast. The second measure $\mu_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.