Name: On a problem of Sárközy and Sós for multivariate linear forms
Keywords: Combinatorial number theory, Àrees temàtiques de la UPC::Matemàtiques i estadística, Combinatorial analysis, Representation function, Representation functions, combinatorial number theory, :11 Number theory [Classificació AMS], :Matemàtiques i estadística [Àrees temàtiques de la UPC], :Matemàtiques i estadística::Matemàtica discreta::Combinatòria [Àrees temàtiques de la UPC], Classificació AMS::11 Number theory

descriptionPublicationkeyboard_double_arrow_right Article 01 Jul 2018 English Publisher:Elsevier BVJournal:Electronic Notes in Discrete Mathematics, volume 68, pages 101-106 (issn: 1571-0653,

Authors: Rué Perna, Juan José; Spiegel, Christoph;

doi: 10.1016/j.endm.2018.06.018 , 10.4171/rmi/1193

handle: 2117/345505 , 2117/121761

On a problem of Sárközy and Sós for multivariate linear forms

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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).

Related Organizations

Barcelona Graduate School of Mathematics
Spain
Universitat Politècnica de Catalunya
Spain
Universitat Polit?cnica de Catalunya
Spain
Universitat Polite`cnica de Catalunya
Spain

Keywords

Combinatorial number theory, Àrees temàtiques de la UPC::Matemàtiques i estadística, Combinatorial analysis, Representation function, Representation functions, combinatorial number theory, :11 Number theory [Classificació AMS], :Matemàtiques i estadística [Àrees temàtiques de la UPC], :Matemàtiques i estadística::Matemàtica discreta::Combinatòria [Àrees temàtiques de la UPC], Classificació AMS::11 Number theory, Arithmetic combinatorics; higher degree uniformity, Additive bases, including sumsets, additive basis, Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria, representation function, representation functions, additive combinatorics, Anàlisi combinatòria

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