Hausdorff dimension of cartesian product The problem statement is:
Show that if the Hausdorff dimension of $X$ is $d$ then the Hausdorff dimension of $X\times X$ is $2d$. I was tried to solve it by showing that $diam(A\times A)=\sqrt{2}diam(A)$ for some arbitrary set $A$ but I had no idea where to go from there. How do I continue this problem?
 A: Only the inequality 
$$\dim (X\times X)\geq 2\dim X$$
 is true. It has many generalizations. You will find a nice combinatorial proof in Marstrand The dimension of cartesian product sets, Proc. Cambridge Philos. Soc. 50 (1954) 198–202. Or one using Frostman's lemma in Mattila's book Geometry of sets and measures in Euclidean spaces Theorem 8.10.
To see that the inequality above may be strict look into Kaoru Hatano, Notes on Hausdorff dimensions of Cartesian product sets Hiroshima Math. J. 1 (1971) 17-25. It is proved there that for any numbers $0\leq\alpha\leq m$, $0\leq\beta\leq n$, $m$ and $n$ integers, and $d$ such that 
$$\alpha+\beta\leq d\leq\min\{m+\beta,\alpha+n\}$$
there exist subsets $E_1\subseteq\mathbb{R}^m$ and $E_2\in\mathbb{R}^n$ such that $\dim E_1=\alpha$, $\dim E_2=\beta$ and 
$$\dim(E_1\times E_2)=d$$
so you may take $X=E_1=E_2$ for $m=n=1$, $\alpha=\beta={1\over 2}$ and $d={3\over 2}$.
See also this answer and the discussion on page 89 in the book https://www.cambridge.org/core/books/fractals-in-probability-and-analysis/D8CBD4181FDC20C387E22939DA2F6168#fndtn-information  available at
https://www.math.stonybrook.edu/~bishop/classes/math324.F15/book1Dec15.pdf
