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

Volume 180 / 2017

Csaba Sándor, Quan-Hui Yang Acta Arithmetica 180 (2017), 161-169 MSC: Primary 11B34; Secondary 11B13. DOI: 10.4064/aa161227-27-4 Published online: 30 August 2017


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 $m\le 35$.


  • Csaba SándorDepartment of Stochastics
    Budapest University of
    Technology and Economics
    H-1529 B.O. Box
    Budapest, Hungary
  • Quan-Hui YangSchool of Mathematics and Statistics
    Nanjing University of
    Information Science and Technology
    Nanjing 210044, China

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