Strong measure zero sets in Polish groups

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.

Meager-Additive Sets in Topological Groups

“…We however have to make sure that T k+1 does not get close to another point of Q * f(k) + x. That is why condition (10) is required. Write = T k + k + t k+1 .…”

“…Use this inequality to estimate d (p + x, ) from below for any p ∈ Q * f(k) , p = q 0 . Here condition (10) gets in play: it yields d (p + x, q 0 + x) = d (p, q 0 ) 3ε f(k+1) and thus…”

“…As to sets in groups, let us recall that Galvin–Mycielski–Solovay Theorem has been extended to all locally compact Polish groups by Kysiak [12] and Fremlin [5]. Further extension of the theorem to Polish groups has been proven to be impossible: Hrušák et al [9] and Hrušák and Zapletal [10] found that under the Continuum Hypothesis the Galvin–Mycielski–Solovay Theorem fails for all Polish groups that are not locally compact.…”

“…The inspiration comes from recent results on strong measure zero: Kysiak [12] and Fremlin [5] showed that the Galvin–Mycielski–Solovay Theorem holds in all locally compact Polish groups. Hrušák et al [9] and Hrušák and Zapletal [10] (see also [11]) found that among Polish groups admitting an invariant metric, under the Continuum Hypothesis the Galvin–Mycielski–Solovay Theorem fails for groups that are not locally compact.…”