A Haar meager set that is not strongly Haar meager

TL;DR

This paper constructs a specific example of a Borel set in an abelian Polish group that is Haar meager but not strongly Haar meager, addressing a key open problem in the area.

Contribution

It provides the first known counterexample separating Haar meager and strongly Haar meager sets, resolving Darji's open question.

Findings

Constructed a G_delta set in al in al that is Haar meager but not strongly Haar meager.

Proved that no F_sigma counterexample exists, establishing the optimality of their result.

Abstract

Following Darji, we say that a Borel subset $B$ of an abelian Polish group $G$ is Haar meager if there is a compact metric space $K$ and a continuous function $f : K \to G$ such that the preimage of the translate, $f^{-1}(B+g)$ is meager in $K$ for every $g \in G$. The set $B$ is called strongly Haar meager if there is a compact set $C \subseteq G$ such that $(B+g) \cap C$ is meager in $C$ for every $g \in G$. The main open problem in this area is Darji's question asking whether these two notions are the same. Even though there have been several partial results suggesting a positive answer, in this paper we construct a counterexample. More specifically, we construct a $G_\delta$ set in $\mathbb{Z}^\omega$ that is Haar meager but not strongly Haar meager. We also show that no $F_\sigma$ counterexample exists, hence our result is optimal.