JournalsrmiVol. 36, No. 7pp. 2107–2119

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
On a problem of Sárközy and Sós for multivariate linear forms cover
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Abstract

We prove that for pairwise co-prime numbers k1,,kd2k_1,\dots,k_d \geq 2 there does not exist any infinite set of positive integers A\mathcal{A} such that the representation function rA(n)=#{(a1,,ad)Ad:k1a1++kdad=n}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 nn 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