Coverings and dimensions in infinite profinite groups

Peter Maga 1
  • 1 Department of Mathematics and its Applications, Central European University, Nádor Str. 9, Budapest, Hungary

Abstract

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.

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