## Ondřej Zindulka

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

Comment.Math.Univ.Carolin. 60,1 (2019) 131-155.**Abstract:**We develop a theory of sharp measure zero sets that parallels Borel's strong measure zero, and prove a theorem analogous to Galvin--Mycielski--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 $f\colon 2^{\omega}\to 2^{\omega}$ is continuous and $X$ is meager-additive, then so is $f(X)$.

**Keywords:** meager-additive; $\mathcal E$-additive; strong measure zero; sharp measure zero; Hausdorff dimension; Hausdorff measure

**DOI:** DOI 10.14712/1213-7243.2015.277

**AMS Subject Classification:** 03E05 03E20 28A78

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