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

TL;DR

This paper extends previous results on the irregularity of additive representation functions by exploring more general conditions and properties related to the sequence of integers and their difference operators.

Contribution

It generalizes earlier theorems about the unboundedness of differences of the representation function for broader classes of sequences.

Findings

Unboundedness of difference operators for generalized sequences

Extension of previous results to higher order differences

Broader conditions under which regularity properties fail

Abstract

Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ 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. Erd\H{o}s, A. S\'ark\"ozy and V. T. S\'os 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.