Note on a problem of S\'ark\"ozy on multiplicative representation functions

Motivated by a 2001 problem of S\'ark\"ozy, we classify all situations of the integers $b,c,e$ and $f$ satisfying \begin{align*} \limsup_{n\rightarrow\infty}|d(\mathcal{A},bn+c)-d(\mathcal{A},en+f)|=\infty \end{align*} for any infinite $\mathcal{A}\subset \mathbb{N}$, where $d(\mathcal{A},m)=\#\{a\in \mathcal{A}:a|m\}.$