多元线性形式的Sárközy和Sós问题

Q2 Mathematics Electronic Notes in Discrete Mathematics Pub Date : 2018-07-01 DOI:10.1016/j.endm.2018.06.018

Juanjo Rué , Christoph Spiegel

引用次数: 5

摘要

证明了对于成对协素数k1，…，kd≥2，不存在任何正整数的无限集A，使得表示函数rA(n)=#{(a1，…，ad)∈ad:k1a1+…+kdad=n}在n足够大时成为常数。这个结果是我们的主要定理的一个特殊情况，它为回答Sárközy和Sós的问题提出了进一步的步骤，并广泛地扩展了Cilleruelo和ru关于二元线性形式的先前结果。伦敦数学学院。社会2009)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On a problem of Sárközy and Sós for multivariate linear forms

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 $A$ such that the representation function $r_{A} (n) = # {(a_{1}, \dots, a_{d}) \in A^{d} : k_{1} a_{1} + \dots + k_{d} a_{d} = n}$ 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).

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Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

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1.30

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0.00%

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期刊介绍： Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.

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