Some extensions of a property of linear representation functions (Q1045025)

Let \(A\) be a set of nonnegative integers. For a fixed \(k \geq 2\) integer, let \(R_{1}(A, n, k)\) denote the number of solutions of the equation \(a_{1} + \dots{} + a_{k} = n\), \(a_{1}, \dots{} ,a_{k} \in A\), and let \(R_{2}(A, n, k)\) and \(R_{3}(A, n, k)\) denote the number of solutions with the additional restrictions \(a_{1} < \dots{} < a_{k}\) and \(a_{1} \leq \dots{} \leq a_{k}\) respectively. In [Acta Math. Hung. 115, No. 1--2, 169--175 (2007; Zbl 1136.11008)], \textit{G. Horváth} proved that if \(d > 0\) is an integer, then there does not exist an \(n_{0}\) such that \(d \leq R_{2}(A, n, 2) \leq d + [\sqrt{2d} + 1/2]\) for \(n > n_{0}\). In this paper the authors extended Horváth's result to any \(k > 2\) for the functions \(R_{1}(A, n, k)\), \(R_{2}(A, n, k)\) and \(R_{3}(A, n, k)\) by using analytic tools.