A question of S\'{a}rkozy and S\'{o}s on representation functions] {A question of S\'{a}rkozy and S\'{o}s on representation functions

For $m\ge 1$, let $0<b_0<b_1<\mathrm{.}\mathrm{.}\mathrm{.}<b_m$ and $e_0,e_1,\mathrm{.}\mathrm{.}\mathrm{.},e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but $p^t\nmid b_i\mathrm{f}\mathrm{o}\mathrm{r}1\le i\le m$. Then, we prove that there is no infinite subset $\mathcal{A}$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+\mathrm{.}\mathrm{.}\mathrm{.}+a_{0,e_0})+\mathrm{.}\mathrm{.}\mathrm{.}+b_m(a_{m,1}+\mathrm{.}\mathrm{.}\mathrm{.}+a_{m,r_m}),a_{i,j}\in \mathcal{A}$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Ru'{e} for bilinear case, and answers a question posed by S'{a}rkozy and S'{o}s.