Meager-additive sets in topological groups

DOI10.1017/JSL.2021.79zbMATH Open1529.22002arXiv1806.06674MaRDI QIDQ6303152

Ondřej Zindulka

Publication date: 8 June 2018

Abstract: By the Galvin-Mycielski-Solovay theorem, a subset $X$ of the line has Borel's strong measure zero if and only if $M+XeqmathbbR$ for each meager set $M$. A set $XsubseteqmathbbR$ is meager-additive if $M+X$ is meager for each meager set $M$. Recently a theorem on meager-additive sets that perfectly parallels the Galvin-Mycielski-Solovay theorem was proven: A set $XsubseteqmathbbR$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero. We investigate the validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants.

Mathematics Subject Classification ID

Cardinal characteristics of the continuum (03E17) General properties and structure of LCA groups (22B05) Analysis on general topological groups (22A10)

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