Haar meager sets, their hulls, and relationship to compact sets

TL;DR

This paper explores the properties of Haar meager sets in abelian Polish groups, demonstrating the existence of complex examples without simple hulls and examining their relationship to compact sets.

Contribution

It introduces new examples of Haar meager sets lacking simple Borel or analytic hulls and analyzes their connection to compact sets in non-locally compact groups.

Findings

Existence of strongly Haar meager sets without $F_{\sigma}$ Haar meager hulls

Existence of coanalytic naively strongly Haar meager sets without Haar meager hulls

Analysis of the relationship between compact sets and Haar meager sets in non-locally compact groups

Abstract

Let $G$ be an abelian Polish group. We show that there is a strongly Haar meager set in $G$ without any $F_{\sigma}$ Haar meager hull (and that this still remains true if we replace $F_{\sigma}$ by any other class of the Borel hierarchy). We also prove that there is a coanalytic naively strongly Haar meager set without any Haar meager hull. Further, we investigate the relationship of the collection of all compact sets to the collection of all Haar meager sets in non-locally compact Polish groups.