2018/06/08 by Zindulka, Ondrej
#03E05 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1806.06674
By the Galvin-Mycielski-Solovay theorem, a subset X of the line has Borel's strong measure zero if and only if M+X≠ℝ for each meager set M. A set X⊆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⊆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.