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Revista Matemática Iberoamericana


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Volume 36, Issue 7, 2020, pp. 2107–2119
DOI: 10.4171/rmi/1193

Published online: 2020-03-18

On a problem of Sárközy and Sós for multivariate linear forms

Juanjo Rué[1] and Christoph Spiegel[2]

(1) Universitat Politècnica de Catalunya, Barcelona, Spain
(2) Universitat Politècnica de Catalunya, Barcelona, Spain

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).

Keywords: Combinatorial number theory, representation function

Rué Juanjo, Spiegel Christoph: On a problem of Sárközy and Sós for multivariate linear forms. Rev. Mat. Iberoam. 36 (2020), 2107-2119. doi: 10.4171/rmi/1193