A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture

Csaba Sándor¹ ; Quan-Hui Yang²

¹ Department of Stochastics Budapest University of Technology and Economics H-1529 B.O. Box Budapest, Hungary
² School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing 210044, China

Acta Arithmetica, Tome 180 (2017) no. 2, pp. 161-169

Résumé

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 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 $m$. 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 $m\le 35$.

Détail

DOI : 10.4064/aa161227-27-4

Keywords: set subseteq mathbb mathbb denote number ordered pairs times celebrated erd tur conjecture says sufficiently large integers nbsp representation function cannot bounded positive integer nbsp ruzsa number defined least positive integer there exists set subseteq mathbb mathbb nbsp chen proved positive integers nbsp paper prove integers determine values