On a problem of Sárközy and Sós for multivariate linear forms

• Juanjo Rué ^[1] ; Christoph Spiegel ^[1]
  1. [1] Universitat Politècnica de Catalunya

     Universitat Politècnica de Catalunya

     Barcelona, España

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 7, 2020, págs. 2107-2119
• Idioma: inglés
• DOI: 10.4171/rmi/1193
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• Resumen

  • We prove that for pairwise co-prime numbers k1,…,kd≥2 there does not exist any infinite set of positive integers A such that the representation function rA(n)=#{(a1,…,ad)∈Ad:k1a1+⋯+kdad=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).