Strong measure zero and meager-additive sets through the prism of fractal measures

TL;DR

This paper introduces the concept of sharp measure zero sets, establishing their equivalence to meager-additive sets, and explores their properties through the lens of fractal measures, extending classical measure theory results.

Contribution

It develops a theory of sharp measure zero sets, proving they are equivalent to meager-additive sets and extending classical theorems to this new context.

Findings

Sharp measure zero sets are equivalent to meager-additive sets.

A subset of 2^ω is meager-additive iff it is E-additive.

Meager-additive sets are preserved under continuous functions.

Abstract

We develop a theory of \emph{sharp measure zero} sets that parallels Borel's \emph{strong measure zero}, and prove a theorem analogous to Galvin-Myscielski-Solovay Theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^\omega$ is meager-additive if and only if it is $\mathcal E$-additive; if $f:2^\omega\to2^\omega$ is continuous and $X$ is meager-additive, then so is $f(X)$.