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

Abstract

For $m\geq 1$, let $0<b_0<b_1<...<b_m$ and $\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\mid b_0$, but $p^t\nmid b_{i}\ {\rm for}\ 1\leq i\leq m$. Then, we prove that there is no infinite subset $\mathcal A$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+...+a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),\ a_{i,j}\in \mathcal A$$ 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.