On a problem of Sárközy and Sós for multivariate linear forms

Abstract

We prove that for pairwise co-prime numbers k1,...,kd = 2 there does not exist any infinite set of positive integers A such that the representation function rA(n) = #{(a1,...,ad) ¿ Ad : k1a1 + ... + kdad = 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 S´ark¨ozy and S´os and widely extends a previous result of Cilleruelo and Ru´e for bivariate linear forms (Bull. of the London Math. Society 2009).