On some properties of representation functions related to the Erdős-Turán conjecture

Abstract

For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a^{\prime })\in A\times A$ such that $a+a^{\prime }=n$. The celebrated Erdős-Turán conjecture says that, if $R_A(n)\ge 1$ for all sufficiently large integers $n$, then the representation function $R_A(n)$ cannot be bounded. For any positive integer $m$, Ruzsa's number $R_m$ is defined to be the least positive integer $r$ such that there exists a set $A\subseteq {\mathbb{Z}}_m$ with $1\le R_A(n)\le r$ for all $n\in {\mathbb{Z}}_m$. In 2008, Chen proved that $R_m\le 288$ for all positive integers $m$. Recently the authors proved that $R_m\ge 6$ for all integers $m\ge 36$. In this paper, we prove that if $A\subseteq {\mathbb{Z}}_m$ satisfies $R_A(n)\le 5$ for all $n\in {\mathbb{Z}}_m$, then $|\{g:g\in {\mathbb{Z}}_m,R_A(g)=0\}|\ge \frac{1}{4}m-\sqrt{5m}$. This improves a recent result of Li and Chen. We also give upper bounds of $|\{g:g\in {\mathbb{Z}}_m,R_A(g)=i\}|$ for $i=2,4$.

9 pages