On a problem of Sárközy and Sós for multivariate linear forms (Q5916271)

Summary: We prove that for pairwise co-prime numbers \(k_1, \ldots, k_d \ge 2\) there does not exist any infinite set of positive integers \(\mathcal{A}\) such that the representation function \(r_{\mathcal{A}}(n) = \# \{(a_1, \ldots, a_d) \in \mathcal{A}^d : k_1 a_1 + \ldots + k_d a_d = n \}\) 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 \textit{A. Sárközy} and \textit{V. T. Sós} [Algorithms Comb. 13, 129--150 (1997; Zbl 0877.11008)]. and widely extends a previous result of \textit{J. Cilleruelo} and the first author for bivariate linear forms [Bull. Lond. Math. Soc. 41, No. 2, 274--280 (2009; Zbl 1218.11014)]. For the preliminary version see Electron. Notes Discrete Math. 68, 101--106 (2018; Zbl 1437.11022).