Thick Difference Sets of Haar Null Compact Sets in Locally Compact Groups

TL;DR

The paper constructs a Haar null compact set in a non-discrete locally compact group whose difference set contains a neighborhood of the identity, and this set can be chosen within any neighborhood.

Contribution

It demonstrates the existence of Haar null compact sets with difference sets containing neighborhoods, constructed within any prescribed neighborhood of the identity.

Findings

Existence of Haar null compact sets with difference sets containing neighborhoods.

Construction of such sets within any prescribed neighborhood.

Difference sets of these sets contain a neighborhood of the identity.

Abstract

Let \(G\) be a non-discrete, locally compact group with Haar measure \(m\). We prove that there exists a compact set \(K \subset G\) with \(m(K)=0\) such that \(KK^{-1}\) contains a neighborhood of the identity. Moreover, such a set may be constructed inside any prescribed neighborhood of the identity.