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On a problem of Sárközy and Sós for multivariate linear forms
Revista Matemática Iberoamericana ( IF 1.3 ) Pub Date : 2020-03-18 , DOI: 10.4171/rmi/1193
Juanjo Rué 1 , Christoph Spiegel 1
Affiliation  

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).

中文翻译:

关于多元线性形式的Sárközy和Sós问题

我们证明对于成对的互质数$ k_1,\ dots,k_d \ geq 2 $不存在任何无穷个正整数$ \ mathcal {A} $,这样表示函数$ r _ {\ mathcal {A} }(n)= \#\ {(a_1,\ dots,a_d){\ in} \ mathcal {A} ^ d:k_1 a_1 + \ cdots + k_d a_d = n \} $对于足够大的$ n $变为常数。这个结果是我们的主要定理的一个特例,它向回答Sárközy和Sós的问题又迈出了一步,并广泛推广了Cilleruelo和Rué先前关于二元线性形式的结果(伦敦数学协会,2009年)。 。
更新日期:2020-03-18
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