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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 $X$.
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    $\begingroup$ Ever heard of the search button? ;-) math.stackexchange.com/questions/580978/… Also, if you know an example with different $X,Y$ of the same dimension, just take $X\cup Y$. $\endgroup$
    – fedja
    Commented Nov 16, 2017 at 23:39
  • $\begingroup$ I searched that question before asking my question. It did not really help me, but yours did. $\endgroup$
    – Thomas Yang
    Commented Nov 17, 2017 at 1:32
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    $\begingroup$ For a simple example, see Theorem 5.11 in Falconer's The Geometry of Fractal Sets $\endgroup$
    – Piotr Hajlasz
    Commented Mar 17, 2018 at 20:42

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