# On a problem of Sárközy and Sós for multivariate linear forms

### Juanjo Rué

Universitat Politècnica de Catalunya, Barcelona, Spain### Christoph Spiegel

Universitat Politècnica de Catalunya, Barcelona, Spain

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## 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 $\mathcal{A}$ 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).

## Cite this article

Juanjo Rué, Christoph Spiegel, On a problem of Sárközy and Sós for multivariate linear forms. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2107–2119

DOI 10.4171/RMI/1193