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：

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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）。 。