Let $A\subset \mathbb R$. Is it true that
$$
\dim(A+A)\le 2\dim A
$$


for some dimensions – say, lower box for the LHS and upper box for the RHS.



Dimension of sumset vs sum of dimensions
$A+A$ is a Lipschitz image of the set $A\times A\subset \mathbb{R}^2$ under the map $(x,y)\to x+y$. If $A$ is covered by $N$ balls of radius $\varepsilon$, then $A\times A$ is covered by $N^2$ balls of radius, say, $\sqrt{2}\varepsilon$, thus the box dimension (lower or upper) of $A\times A$ does not exceed twice that of $A$, and the Lipschitz map does not increase the dimension.


