  The Hausdorff dimension of a product XxY 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 XxY 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 XxY are the same.
  We give a negative answer: Assuming CH, there exists a gamma-set of reals X
and a set of reals Y with Hausdorff dimension zero, such that the Hausdorff
dimension of X+Y (a Lipschitz image of XxY) 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.
