On a problem of S\'ark\"ozy and S\'os for multivariate linear forms

TL;DR

The paper proves that for certain multivariate linear forms with pairwise coprime coefficients, no infinite set of positive integers can have a constant representation function for large n, advancing understanding of additive number theory.

Contribution

It establishes a new non-existence result for infinite sets with constant representation functions in multivariate linear forms, extending previous bivariate results.

Findings

No infinite set A has a constant representation function for large n in the specified forms.

The result generalizes earlier bivariate cases to higher dimensions.

It advances the understanding of additive representations in number theory.

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 $A$ such that the representation function $r_A (n) = \{ (a_1, \dots, a_d) \in A^d : k_1 a_1 + \dots + k_d a_d = n \}$ 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 S\'ark\"ozy and S\'os and widely extends a previous result of Cilleruelo and Ru\'e for bivariate linear forms.