A Haar meager set that is not strongly Haar meager

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.