Given a positive integer $q$ and a finite set $A\subseteq \mathbb{N}$, define the functions $$ f(q)=\sum_{k=1}^{\infty}\sum_{\substack{n,m\in A \\ n-m=qk}}1 \quad \mbox{and}\quad g(q)=\sum_{\substack{n,m\in A \\ q|n-m}}1. $$ I want to know if there any relations between the two functions. I think for instance it is true that $f(q)\le g(q)$, but I would like more explicit relations, maybe something like $f(q)=c g(q)$ where we can compute $c$ explicitely or at least bound it.
An example is for instance $A=\{1,2\ldots, 9\}$ and $q=1$. Then, $f(1)=36$ and $g(1)=81$. In this case $c=4/9 \lt 1/2$.