Locally Random Groups

TL;DR

This paper introduces the concept of local randomness in compact metric groups, establishing mixing inequalities, product results, and entropy gain properties, with applications to spectral gaps and new examples of such groups.

Contribution

It develops the theory of local randomness in compact groups, proving key inequalities and results, and connects these to spectral gaps and entropy in a novel way.

Findings

Proved a mixing inequality for locally random groups.

Established a product result under a dimension condition.

Developed a Littlewood-Paley decomposition and linked it to spectral gaps.

Abstract

In this work, we will introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we will develop a Littlewood-Paley decomposition and explore its connection to the existence of the spectral gap for random walks. Moreover, under the dimension condition alone, we will prove a multi-scale entropy gain result `a la Bourgain-Gamburd and Tao.