A lower bound of Ruzsa's number related to the Erd\H{o}s-Tur\'an conjecture

TL;DR

This paper investigates Ruzsa's number related to the Erdős-Turán conjecture, establishing new bounds and exact values for Ruzsa's number for various moduli, advancing understanding of additive representation functions.

Contribution

It proves that Ruzsa's number is at least 6 for all integers greater than or equal to 36 and determines all Ruzsa's numbers for moduli up to 35, providing new bounds and exact values.

Findings

Ruzsa's number R_m ≥ 6 for all m ≥ 36

Exact Ruzsa's numbers determined for m ≤ 35

Improved bounds on additive representation functions

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\'{a}n 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 $m$. In this paper, we prove that $R_m\ge 6$ for all integers $m\ge 36$. We also determine all values of $R_m$ when $m\le 35$.