On an additive property of sequences of nonnegative integers
Let \(A=\{a_1\leq a_2\leq \dots\}\) be a sequence of natural numbers and denote by \(R_k(n)\) the number of solutions of \(n=\sum_{j=1}^ka_{i_j}\). In the middle of the eighties \textit{P. Erdős} and \textit{A. Sárközy} [Pac. J. Math. 118, 347-357 (1985; Zbl 0569.10032)] proved that if \(A=\{0<a_1<a_2< \dots\}\) is a strictly increasing sequence of integers, \(F(n)\) is an arithmetic function satisfying \(F(n)\longrightarrow \infty\), \(F(n)\leq F(n+1)\) for \(n\geq n_0\) and \(F(n)=o(n(\log n)^{-2})\), then \(|R_2(n)-F(n)|=o(\sqrt{F(n)})\) cannot hold. In the present paper the author extends this result for \(k>2\), proving that if the above mentioned conditions hold for \(F(n)\), then \(\sum_{n=1}^N(R_k(n)-F(n))^2=o(NF(N))\) cannot hold. The proof is based on the original proof of Erdős and Sárközy.
scientific article; zbMATH DE number 2016839