Using the definition of the Hausdorff measure as
$$H^\alpha(C) = \lim_{\delta\rightarrow\infty} H_\delta^\alpha(C) = \lim_{\delta\rightarrow\infty} \left( \inf\sum_{i=1}^\infty |\operatorname{diam}(U_i)|^\alpha \right)$$
where the $\alpha$ is the Hausdorff dimension of $C$ and the infimum is taken over all sets $U_i$ such that the union covers $C,$ and $\operatorname{diam}(U_i) \leq \delta$
Can I say for the Cartesian product $C \times C$ that
$$H^{\alpha+\alpha}(C \times C) \geq H^\alpha(C)H^\alpha(C)$$