I am thinking of the following question:

Let $X\subseteq \mathbb R$. Is it true that $$ \mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)? $$

My thoughts:

1. We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(Y)\leq \mathrm{dim_H}(X\times Y)\leq \mathrm{dim_H}(X)+\mathrm{dim_M}(Y)$ for all metric spaces $X,Y$, where $\mathrm{dim_M}$ denotes the upper Minkowski dimension. Therefore if $\mathrm{dim_H}(X)=\mathrm{dim_M}(X)$, the above is always true.
2. All counterexamples to the equation $\mathrm{dim_H}(X)+\mathrm{dim_H}(Y)= \mathrm{dim_H}(X\times Y)$ as far as I know use different $X$ and $Y$'s. Is there a counterexample with the same $X$?
3. It is again trivial if $X$ is countable. So we are mainly interested in an uncountable $X$.