Strong measure zero in Polish groups

TL;DR

This paper explores the concept of strong measure zero in Polish groups, examining its properties, relationships with Hausdorff measure, and behavior under products, extending classical theorems to a broader context.

Contribution

It extends the classical theorem of Galvin, Mycielski, and Solovay to arbitrary Polish groups and introduces the notion of sharp measure zero, linking it to meager-additive sets.

Findings

Hausdorff measure characterizes strong measure zero

Products of strong measure zero sets are analyzed

Sharp measure zero relates to meager-additive sets

Abstract

The notion of strong measure zero is studied in the context of Polish groups. In particular, the extent to which the theorem of Galvin, Mycielski and Solovay holds in the context of an arbitrary Polish group is studied. Hausdorff measure and dimension is used to characterize strong measure zero. The products of strong measure zero sets are examined. Sharp measure zero, a notion stronger that strong measure zero, is shown to be related to meageradditive sets in the Cantor set and Polish groups by a theorem very similar to the theorem of Galvin, Mycielski and Solovay.