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MEAGER-ADDITIVE SETS IN TOPOLOGICAL GROUPS

Part of: Topological and differentiable algebraic systems Locally compact abelian groups Set theory

Published online by Cambridge University Press:  27 September 2021

ONDŘEJ ZINDULKA
ONDŘEJ ZINDULKA*
Affiliation:
DEPARTMENT OF MATHEMATICS FACULTY OF CIVIL ENGINEERING CZECH TECHNICAL UNIVERSITY THÁKUROVA 7, 160 00 PRAGUE 6, CZECH REPUBLICE-mail:ondrej.zindulka@cvut.czURL: http://mat.fsv.cvut.cz/zindulka
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Abstract

By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M.

A set $X\subseteq \mathbb {R}$ 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 $X\subseteq \mathbb {R}$ 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.

Keywords

Polish groupmeager-additivesharply meager-additivesharp measure zerostrong measure zeroHausdorff measure

MSC classification

Primary: 22B05: General properties and structure of LCA groups
Secondary: 22A10: Analysis on general topological groups 03E17: Cardinal characteristics of the continuum
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 1046 - 1064
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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