Topological diagonalizations and Hausdorff dimension

Abstract

The Hausdorff dimension of a product $X\times Y$ can be strictly greater than that of $Y$, even when the Hausdorff dimension of $X$ is zero. But when $X$ is countable, the Hausdorff dimensions of $Y$ and $X\times Y$ are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than "being countable" and stronger than "having Hausdorff dimension zero". Fremlin asked whether it is enough for $X$ to have the strongest property in this hierarchy (namely, being a $\gamma$-set) in order to assure that the Hausdorff dimensions of $Y$ and $X\times Y$ are the same.  
We give a negative answer: Assuming the Continuum Hypothesis, there exists a $\gamma$-set $X \subseteq \mathbb{R}$ and a set $Y \subseteq \mathbb{R}$ with Hausdorff dimension zero, such that the Hausdorff dimension of $X+Y$ (a Lipschitz image of $X\times Y$) is maximal, that is, $1$. However, we show that for the notion of a strong $\gamma$-set the answer is positive. Some related problems remain open.