Let $f,g\in{\mathcal C}[0,1]$ (continuous real-valued funtions defined on $[0,1]$).
What can be said about the (Hausdorff or box-counting) dimension of the graph of $f\cdot g$ in terms of the dimensions of the graphs of $f$ and $g$?
In particular, what can be said about the dimension of the graph of $f^2$ in terms of the dimension of the graph of $f$?
Here is a solution to the second question. Let $I(f)$ and $I(f^2)$ be the graphs of $f$ and $f^2$. For $n\in\mathbb Z$, let $A_n:=f^{-1}([2^n,2^{n+1}))$, $B_n:=f^{-1}((-2^{n+1},-2^n])$ and $C:=f^{-1}(0)$. It is easy to show that the map $(x,f(x))\mapsto (x,f^2(x))$ is bi-Lipschitz on each of the sets $A_n$, $B_n$ and $C$. So the Hausdorff dimension of $I(f)\cap (A_n\times \mathbb R)$ is equal to the Hausdorff dimension of $I(f^2)\cap (A_n\times \mathbb R)$, etc. It follows that $I(f)$ and $I(f^2)$ have the same Hausdorff dimension.
I don't know if this is true for the Minkowski dimension or not.