Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

TL;DR

This paper explores the similarities between Haar meager and Haar null sets in abelian Polish groups, establishing new properties, examples, and a topological analog of Christensen measurability to analyze their structure and implications.

Contribution

It introduces the concept of D-measurability as a topological analog of Christensen measurability and applies a generalized Piccard's theorem to prove the continuity of D-measurable homomorphisms.

Findings

0 is in the interior of A-A for each non-Haar meager Borel set A

Constructs a Borel non-Haar meager set A with empty interior of A+A in c_0

Proves that D-measurable homomorphisms are continuous

Abstract

In the paper we would like to pay attention to some analogies between Haar meager sets and Haar null sets. Among others, we will show that $0\in \inn (A-A)$ for each Borel set $A$, which is not Haar meager in an abelian Polish group. Moreover, we will give an example of a Borel non-Haar meager set $A\subset c_0$ such that $\inn (A+A)=\emptyset$. Finally, we will define $D$-measurability as a topological analog of Christensen measurability, and apply our generalization of Piccard's theorem to prove that each $D$-measurable homomorphism is continuous. Our results refer to the papers \cite{Ch}, \cite{Darji} and \cite{FS}.