On a question of Sárkozy and Sós for bilinear forms

Abstract

We prove that if 2 ≤ k1 ≤ k2, then there is no infinite sequence $\emph{A}$ of positive integers such that the representation function r(n)=#{(a, a'): n=$k{_1}a$ + $k{_2}a'$, a,a' ∊ $\emph{A}$} is constant for n large enough. This result completes previous work of Dirac and Moser for the special case $k_1$ = 1 and answers a question posed by Sárkozy and Sós.