Topological diagonalizations and Hausdorff dimension

The Hausdorff dimension of a product X × 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 × 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 γ-set) in order to assure that the Hausdorff dimensions of Y and X × Y are the same. We give a negative answer: Assuming the Continuum Hypothesis, there exists a γ-set X ⊆ R and a set Y ⊆ R with Hausdorff dimension zero, such that the Hausdorff dimension of X + Y (a Lipschitz image of X × Y ) is maximal, that is, 1. However, we show that for the notion of a strong γ-set the answer is positive. Some related problems remain open.