Suppose that I have two asymptotic counts given by $$ \#\{x \in [0,H] \cap \mathbb Z: f(x) \leq H\} \sim F(H) $$ and also $$ \#\{x \in [0,H] \cap \mathbb Z: g(x) \leq H\} \sim G(H). $$

From these two counts, is there any way to deduce the asymptotic growth of $$ \#\{x \in [0,H] \cap \mathbb Z: f(x) + g(x) \leq H\}? $$

It seems that there are some ways to approximate this if $f, g$ are arithmetic functions, and in theory it seems to me that the answer should in some way be related to the convolution of $F(H)$ and $G(H)$, but I don't readily see it.