# On some properties of representation functions related to the Erd\H{o}s-Tur\'an conjecture

## TL;DR

This paper investigates properties of representation functions related to the Erdős-Turán conjecture, providing bounds on the number of elements with zero representation and bounds for certain representation counts in modular arithmetic.

## Contribution

It establishes new lower bounds on the number of elements with zero representation when the representation function is bounded by 5, improving previous results.

## Key findings

- If R_A(n) ≤ 5, then at least a quarter of the elements have zero representation.
- Provides upper bounds for the number of elements with representation counts 2 and 4.
- Improves bounds related to the Erdős-Turán conjecture in modular settings.

## Abstract

For a set $A\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. The celebrated Erd\H{o}s-Tur\'an 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 $i=2,4$.