A Haar meager set that is not strongly Haar meager

Elekes, Márton and Nagy, Donát and Poór, Márk and Vidnyánszky, Zoltán (2020) A Haar meager set that is not strongly Haar meager. ISRAEL JOURNAL OF MATHEMATICS, 235 (1). pp. 91-109. ISSN 0021-2172

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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 → 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 Gδ set in ℤω that is Haar meager but not strongly Haar meager. We also show that no Fσ counterexample exists, hence our result is optimal. © 2019, The Hebrew University of Jerusalem.

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