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 subset of positive integers can produce a constant number of solutions for a specific linear equation, generalizing previous results and answering a posed question.

Contribution

It extends prior work by showing the non-existence of infinite sets with constant solution counts for a class of linear equations under divisibility constraints.

Findings

No infinite subset of positive integers yields constant solutions for the given equation.

Generalizes previous bilinear case results to more complex linear equations.

Provides an answer to a question posed by Sárkozy 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 $$n=b_0(a_{0,1}+\cdot +a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),\ a_{i,j}\in \mathcal A$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Ru\'{e} for the bilinear case, and answers a question posed by S\'{a}rkozy and S\'{o}s.