On a problem of Sárközy and Sós for multivariate linear forms
Abstract
We prove that for pairwise coprime 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 DOI:
 10.48550/arXiv.1802.07597
 arXiv:
 arXiv:1802.07597
 Bibcode:
 2018arXiv180207597R
 Keywords:

 Mathematics  Combinatorics;
 11B75;
 11B13;
 11B34
 EPrint:
 Added clarifications regarding the particular notion of limit used in the first part of the paper. 11 pages