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)。

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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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来源期刊

Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

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