Measure of the Cantor set multiplied by the Cantor set
Your question is already treated in mathoverflow. See Arithmetic products of Cantor Sets with interesting, nearly up-to-date information. The question treats self-similar sets on $\mathbb R$ in general. Results with respect to Cantor sets are e.g. that the Hausdorff Dimension $$\dim_H(C\cdot C) = \min(2 \cdot \dim_H(C),1)=\min\left(2\frac{\ln 2}{\ln 3},1\right) = 1.$$ Another interesting aspect is, that according to the experts there, an answer to the question if $C\cdot C$ has positive measure seems to be out of reach at the time.