A Haar meager set that is not strongly Haar meager

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 → G such that the preimage of the translate f −1 ( B + g ) is meager in K for every g ∈ G . The set B is called strongly Haar meager if there is a compact set C ⊆ G such that ( B + g ) ⋂ C is meager in C for every g ∈ 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 δ set in ℤ ω that is Haar meager but not strongly Haar meager. We also show that no F σ counterexample exists, hence our result is optimal.