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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 continuousfunction $f : K \to G$ 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 Haarmeager if there is a compact set $C \subseteq G$ such that $(B+g) \cap C$ ismeager in $C$ for every $g \in G$. The main open problem in this area isDarji's question asking whether these two notions are the same. Even thoughthere have been several partial results suggesting a positive answer, in thispaper we construct a counterexample. More specifically, we construct a$G_\delta$ set in $\mathbb{Z}^\omega$ that is Haar meager but not strongly Haarmeager. We also show that no $F_\sigma$ counterexample exists, hence our resultis optimal.

A Haar meager set that is not strongly Haar meager