On a problem of Sárközy and Sós for multivariate linear forms

Abstract

We prove that for pairwise co-prime numbers $k_1,\dots ,k_d\ge 2$ there does not exist any infinite set of positive integers $\mathcal{A}$ such that the representation function $r_{\mathcal{A}}(n)=\#\{(a_1,\dots ,a_d)\in {\mathcal{A}}^d:k_1{a}_1+\cdots +k_d{a}_d=n\}$ becomes constant for $n$ large enough. This result is a particular case of our main theorem, which poses a further step towards answering a question of Sárközy and Sós and widely extends a previous result of Cilleruelo and Rué for bivariate linear forms (Bull. of the London Math. Society, 2009).