MEAGER-ADDITIVE SETS IN TOPOLOGICAL GROUPS

By theGalvin-Mycielski-Solovay theorem, a subset X of the line has Borel's strong measure zero if and only if M + X not equal R for each meager set M. A set X subset of R 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 X subset of R 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.