# On the values of representation functions II

## TL;DR

This paper investigates the asymptotic behavior of representation functions for subsets of nonnegative integers, establishing conditions under which these functions approximate linear growth for almost all integers, thus advancing previous results.

## Contribution

It proves new conditions ensuring that the representation functions approximate linear functions for almost all integers, improving earlier findings by the first author.

## Key findings

- Representation functions approximate n/8 for almost all n
- Conditions on A ensure asymptotic linearity of R_2 and R_3
- Results extend previous work on representation functions

## Abstract

For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$ and $a\leq a'$, respectively. In this paper, we prove that, if $A\subseteq \mathbb{N}$ and $N$ is a positive integer such that $R_2(A,n)=R_2(\mathbb{N}\setminus A,n)$ for all $n\geq2N-1$, then for any $\theta$ with $0<\theta<\frac{2\log2-\log3}{42\log 2-9\log3}$, the set of integers $n$ with $R_2(A,n)=\frac{n}{8}+O(n^{1-\theta})$ has density one. The similar result holds for $R_3(A,n)$. These improve the results of the first author.