A combinatorial proof of Marstrand's Theorem for products of regular Cantor sets.

link: http://arxiv.org/abs/0911.1191
Abstract

In \cite{Ma}, Marstrand proved that if $K\subset\R^2$ has Hausdorff dimension
greater than 1, then its one-dimensional projection has positive Lebesgue
measure for almost-all directions. In this article, we give a combinatorial
proof of this theorem when $K$ is the product of regular Cantor sets of class
$C^{1+a}$, $a>0$, for which the sum of their Hausdorff dimension is greater
than 1.