On a problem of Sárközy and Sós for multivariate linear forms

Abstract

We prove that for pairwise co-prime numbers $k\_1,\dots,k\_d \geq 2$ 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).