A question of Sárkozy and Sós on representation functions

Abstract

For $m\ge 1$, 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 $p^t\nmid b_i\mathrm{for}1\le i\le m$. Then, we prove that there is no infinite subset $\mathcal{A}$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+\cdot +a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),a_{i,j}\in \mathcal{A}$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Rué for the bilinear case, and answers a question posed by Sárkozy and Sós.

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