The closed Steinhaus properties of $\sigma$-ideals on topological groups

TL;DR

This paper investigates the properties of certain $\sigma$-ideals on topological groups, establishing conditions under which meager subgroups belong to these ideals and exploring generalizations and counterexamples in various group contexts.

Contribution

It introduces the closed $ ext{±}n$-Steinhaus property for $\sigma$-ideals, proves meager quasi-analytic subgroups belong to specific ideals, and constructs examples and counterexamples in non-locally compact groups.

Findings

Meager quasi-analytic subgroups belong to $\sigma$-ideals with the closed $ ext{±}n$-Steinhaus property.

The $\sigma$-ideal generated by closed Haar null sets has the closed $ ext{±}2$-Steinhaus property.

Counterexamples show certain meager subgroups are not covered by countably many closed Haar-null sets.

Abstract

We prove that any meager quasi-analytic subgroup of a topological group $G$ belongs to every $\sigma$-ideal $\mathcal I$ on $G$ possessing the closed $\pm n$-Steinhaus property for some $n\in\mathbb N$. An ideal $\mathcal I$ on a topological group $G$ is defined to have the closed $\pm n$-Steinhaus property if for any closed subsets $A_1,\dots,A_n\notin\mathcal I$ of $G$ the product $(A_1\cup A_1^{-1})\cdots (A_n\cup A_n^{-1})$ is not nowhere dense in $G$. Since the $\sigma$-ideal $\mathcal E$ generated by closed Haar null sets in a locally compact group $G$ has the closed $\pm 2$-Steinhaus property, we conclude that each meager quasi-analytic subgroup $H\subset G$ belongs to the ideal $\mathcal E$. For analytic subgroups of the real line this result was proved by Laczkovich in 1998. We shall discuss possible generalizations of the Laczkovich Theorem to non-locally compact groups and construct an example of a meager Borel subgroup in $\mathbb Z^\omega$ which cannot be covered by countably many closed Haar-null (or even closed Haar-meager) sets. On the other hand, assuming that $cof(\mathcal M)=cov(\mathcal M)=cov(\mathcal N)$ we construct a subgroup $H\subset 2^\omega$ which is meager and Haar null but does not belong to the $\sigma$-ideal $\mathcal E$. The construction uses a new cardinal characteristic $voc^*(\mathcal I,\mathcal J)$ which seems to be interesting by its own.