A note on inverse results of random walks in Abelian groups

TL;DR

This paper characterizes when random walks on finite Abelian groups significantly deviate from uniform distribution, revealing near optimal conditions and connecting to existing theoretical results.

Contribution

It provides new near optimal characterizations of the discrepancy of random walks on finite Abelian groups and links these findings to prior research.

Findings

Identifies conditions for large discrepancy in random walks

Establishes near optimal bounds for deviation from uniformity

Connects new characterizations with existing literature

Abstract

In this short note we give various near optimal characterizations of random walks over finite Abelian groups with large maximum discrepancy from the uniform measure. We also provide several interesting connections to existing results in the literature.