Haar-open sets: a right way of generalizing the Steinhaus sum theorem to non-locally compact groups

TL;DR

This paper extends the classical Steinhaus sum theorem to certain non-locally compact groups by proving that sum-sets of non-Haar-null Borel sets are Haar-open, revealing new structural properties in these groups.

Contribution

It introduces the concept of Haar-open sets and proves a Steinhaus-type result for countable products of Abelian locally compact Polish groups, a significant generalization.

Findings

Sum-sets of non-Haar-null Borel sets are Haar-open in the specified groups.

Generalization of Steinhaus Theorem to non-locally compact groups.

Open question about applicability to Banach spaces.

Abstract

Let $X$ be the countable product of Abelian locally compact Polish groups and $A,B\subset X$ be two Borel sets, which are not Haar-null in $X$. We prove that the sum-set $A+B:=\{a+b:a\in A,\;\;b\in B\}$ is Haar-open in the sense that for any non-empty compact subset $K\subset X$ and point $p\in K$ there exists a point $x\in X$ such that the set $K\cap(A+B+x)$ is a neighborhood of $p$ in $K$. This is a generalization of the classical Steinhaus Theorem (1920) to non-locally compact groups. We do not know if this generalization holds for Banach spaces.