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

TL;DR

This paper proves that under certain divisibility conditions, no infinite set of positive integers can produce a constant number of solutions for a specific linear equation, extending previous results and answering a question by Sárközy and Sós.

Contribution

It generalizes prior work by Cilleruelo and Rué to a broader class of equations, resolving an open question by Sárközy and Sós.

Findings

No infinite subset yields constant solutions for the equation under given conditions.

Generalizes previous bilinear case results to more complex linear equations.

Answers an open problem posed by Sárközy and Sós.

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.