On a problem of Sárközy and Sós for multivariate linear forms

Juanjo Rué; Christoph Spiegel

paper

On a problem of Sárközy and Sós for multivariate linear forms

arXiv:1802.07597

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 Sárközy and Sós and widely extends a previous result of Cilleruelo and Rué for bivariate linear forms.

Added clarifications regarding the particular notion of limit used in the first part of the paper. 11 pages