Note on a problem of Sárközy on multiplicative representation functions

Motivated by a 2001 problem of Sárközy, 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 $d(\mathcal{A},m)=\mathrm{\#}\{a\in \mathcal{A}:a\mathrm{\mid }m\}\mathrm{.}$