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

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## 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.

## Key words and phrases

additive number theory general sequence additive representation function## Mathematics Subject Classification

primary 11B34## Preview

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