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.
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All four authors were supported by the National Research, Development and Innovation Office - NKFIH, grants no. 104178 and 124749. The first and fourth authors were also supported by the National Research, Development and Innovation Office - NKFIH, grant no. 113047. The fourth author was also supported by FWF Grant P29999.
Supported by the ÚNKP-17-3 New National Excellence Program of the Ministry of Human Capacities.
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Elekes, M., Nagy, D., Poór, M. et al. A Haar meager set that is not strongly Haar meager. Isr. J. Math. 235, 91–109 (2020). https://doi.org/10.1007/s11856-019-1950-y
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DOI: https://doi.org/10.1007/s11856-019-1950-y