Meager-additive sets in topological groups

TL;DR

This paper explores the properties of meager-additive sets in topological groups, extending a known characterization from the real line to more general Polish groups, and investigates their measure-theoretic and cardinal invariants.

Contribution

It generalizes the characterization of meager-additive sets with sharp measure zero from the real line to locally compact Polish groups with invariant metrics.

Findings

Meager-additive sets in locally compact Polish groups are characterized by sharp measure zero.

The paper establishes the equivalence between meager-additivity and sharp measure zero in these groups.

Some cardinal invariants related to these sets are computed.

Abstract

By the Galvin-Mycielski-Solovay theorem, a subset $X$ of the line has Borel's strong measure zero if and only if $M+X\neq\mathbb{R}$ for each meager set $M$. A set $X\subseteq\mathbb{R}$ is meager-additive if $M+X$ is meager for each meager set $M$. Recently a theorem on meager-additive sets that perfectly parallels the Galvin-Mycielski-Solovay theorem was proven: A set $X\subseteq\mathbb{R}$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero. We investigate the validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants.