Non-Expanding Random walks on Homogeneous spaces and Diophantine approximation

TL;DR

This paper investigates non-expanding random walks on affine lattices, establishing a classification of stationary measures, linking genericity concepts, and applying findings to inhomogeneous Diophantine approximation on fractals.

Contribution

It introduces a new classification theorem for stationary measures of non-expanding random walks and connects genericity notions to Diophantine approximation.

Findings

Classification theorem for stationary measures

Relation between genericity and Birkhoff genericity

Applications to Diophantine approximation on fractals

Abstract

We study non-expanding random walks on the space of affine lattices and establish a new classification theorem for stationary measures. Further, we prove a theorem that relates the genericity with respect to these random walks to Birkhoff genericity. Finally, we apply these theorems to obtain several results in inhomogeneous Diophantine approximation, especially on fractals.