An extension of Nathanson's theorem on representation functions
Let \(\mathbb{N}\) be the set of nonnegative integers. For a given infinite set \(A \subset \mathbb{N}\) the representation function, \(R^{1}_{h,A}(n)\) denotes the number of solutions to the equation \(n=a_{i_{1}}+\dots+ a_{i_{h}}, a_{i_{j}}\in A\). Similarly \(R^{2}_{h,A}(n)\)and \(R^{3}_{h,A}(n)\) denote the number of solutions to the equation \(n=a_{i_{1}}+\dots+a_{i_{h}}, a_{i_{j}}\in A\) such that \(a_{i_{1}} \leq \dots\leq a_{i_{h}}\) and \(a_{i_{1}} <\dots< a_{i_{h}}\) respectively. A. Sárközy asked whether there exist two infinite sets of nonnegative integers \(A\) and \(B\) with infinite symmetric difference and \( R^{i}_{2,A}(n)=R^{i}_{2,B}(n)\) if \(n \geq n_{0}\), for \(i = 1, 2, 3\). G. Dombi in 2002 showed that the answer is negative for \(i = 1.\) For \(i = 2 \), \textit{G. Dombi} [Acta Arith. 103, No. 2, 137--146 (2002; Zbl 1014.11009)], and for \(i = 3\), \textit{Y. G. Chen and B. Wang} [Acta Arith. 110, No. 3, 299--303 (2003; Zbl 1032.11008)] proved that the set of nonnegative integers can be partitioned into two subsets \(A\) and \(B\) such that \( R^{i}_{2,A}(n)=R^{i}_{2,B}\) for all \(n \geq n_{0}\). Using generating functions \textit{Cs. Sándor} [Integers 4, Paper A18, 5 p. (2004; Zbl 1135.11305)] determined the sets \(A \subset \mathbb{N}\) for which either \( R^{2}_{2,A}(n)=R^{2}_{2,\mathbb{N}- A}(n)\) for all \(n \geq n_{0}\) or \( R^{3}_{2,A}(n)=R^{3}_{2,\mathbb{N}-A}(n)\) for all \(n \geq n_{0}\). In this paper all pairs \((A,B), A;B \subset \mathbb{N}\) for which \( R^{1}_{h,A}(n)=R^{1}_{h,B}\) from a point on, where \(h\) is a power of a prime, have been determined. The composite case has also been discussed.
scientific article; zbMATH DE number 6837917