Generalizations of some results about the regularity properties of an additive representation function

Abstract

Let ${A = \{a_{1},a_{2},\dots{} \}}$${(a_{1} < a_{2} < \cdots )}$ be an infinite sequence of nonnegative integers, and let ${R_{A,2}(n)}$ denote the number of solutions of ${a_{x}+a_{y}=n}$${(a_{x},a_{y} \in A)}$. P. Erdős, A. Sárközy and V. T. Sós proved that if ${\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty}$ then ${|\Delta_{1}(R_{A,2}(n))|}$ cannot be bounded, where ${B(A,N)}$ denotes the number of blocks formed by consecutive integers in A up to N and ${\Delta_{l}}$ denotes the l-th difference. Their result was extended to ${\Delta_{l}(R_{A,2}(n))}$ for any fixed ${l\ge2}$. In this paper we give further generalizations of this problem.