Haar-smallest sets

TL;DR

This paper investigates various notions of small sets in abelian Polish groups, focusing on their properties, examples, and the structure of related sigma-ideals, including the non-ideal nature of Haar-countable sets.

Contribution

It provides a comprehensive analysis of Haar-small sets, introduces new examples, and addresses open questions about their algebraic and measure-theoretic properties.

Findings

Haar-countable sets do not form an ideal.

Examples distinguish different smallness notions in all abelian Polish groups.

Answers to questions on null-finite sets are provided.

Abstract

In this paper we are interested in the following notions of smallness: a subset $A$ of an abelian Polish group $X$ is called Haar-countable/Haar-finite/Haar-$n$ if there are a Borel hull $B\supseteq A$ and a copy $C$ of $2^\omega$ such that $(C+x)\cap B$ is countable/finite/of cardinality at most $n$, for all $x\in X$. Recently, Banakh et al. have unified the notions of Haar-null and Haar-meager sets by introducing Haar-$\mathcal{I}$ sets, where $\mathcal{I}$ is a collection of subsets of $2^\omega$. It turns out that if $\mathcal{I}$ is the $\sigma$-ideal of countable sets, the ideal of finite sets or the collection of sets of cardinality at most $n$, then we get the above notions. Moreover, those notions have been studied independently by Zakrzewski (under a different name -- perfectly $\kappa$-small sets). We study basic properties of the corresponding families of small sets, give suitable examples distinguishing them (in all abelian Polish groups of the form $\mathbb{R}\times X$) and study $\sigma$-ideals generated by compact members of the considered families. In particular, we show that Haar-countable sets do not form an ideal. Moreover, we answer some questions concerning null-finite sets, asked by Banakh and Jab{\l}o\'nska, and pose several open problems.