Assume $A,B\subseteq \mathbb{R}^n$. Is it true that $\dim_H(A+B)\le \dim_HA+\dim_HB $?
When $\dim_HA+\dim_HB\ge n$, this is trivial.
The line $10$ in the link 1 says that the answer for lower-box dimension is positive. But What about Hausdorff dimension?
Example 7.8 page 97 of Falconer's book Fractal geometry gives an example of two subsets of $\mathbb{R}$ of Hausdorff dimension 0 and whose sum is of Hausdorff dimension 1.