On a regularity property of additive representation functions

Summary

Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathcal{A}=\{a_{1},a_{2},\dots{}\}$ $(a_{1} \le a_{2} \le \dots{})$ be an infinite sequence of nonnegative integers, and let $R(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in\mathcal{A})$. P. Erdõs, A. Sárközyand V. T. Sós proved that if $\lim_{N\to\infty}\frac{B(\mathcal{A},N)}{\sqrt{N}}=+\infty$ then $|\Delta_{1}(R(n))|$ cannot be bounded, where ${B(\mathcal{A},N)}$ denotes the number of blocks formed by consecutive integers in $\mathcal{A}$ up to $N$ and $\Delta_{k}$ denotes the $k$-th difference. The aim of this paper is to extend this result to $\Delta_{k}(R(n))$ for any fixed $k\ge2$.