Solutions to Two Problems of Sárközy and Sós on Additive Representation Functions

Abstract

For a set $A\subseteq {\mathbb{N}}_0$, let $r_1(A,n)$ denote the number of solutions of the equation $a+a^{\prime }=n$ with $a,a^{\prime }\in A$, and let $r_2(A,n)$ denote the number of such solutions subject to $a\le a^{\prime }$. These functions are called additive representation functions (as first considered by Erdős, Sárközy and Sós). In this paper, we resolve two problems posed by Sárközy and Sós in 1997. First, if $A$ is infinite and $r_2(A,2m+1)\ge r_2(A,2m)$ for every sufficiently large $m$, then the complement of $A$ is finite. This gives a negative answer to Problem 3.1 in~\cite{SarkozySos1997}. Secondly, there exist an arithmetic function $f$ satisfying $f(n)\to \infty$, $f(n+1)\ge f(n)$ for $n>n_0$, and $f(n)=o\left(\frac{n}{(\log n{)}^2}\right)$, and a set $A$ such that \( |r_1(A,n) - f(n)| = o((f(n))^{1/2}) \) holds on a sequence of integers $n$ whose density is $1$. This gives a positive answer to Problem 3.3 in~\cite{SarkozySos1997}.