A basis of ${\bf Z}_m$

Min Tang¹ ; Yong-Gao Chen²

¹ Department of Mathematics Anhui Normal University Wuhu 241000, China
² Department of Mathematics Nanjing Normal University Nanjing 210097, China

Colloquium Mathematicum, Tome 104 (2006) no. 1, pp. 99-103

Résumé

Let $\sigma_A(n)=|\{(a,a')\in A^2: a+a'=n\}|$, where $n\in {\bf N}$ and $A$ is a subset of ${\bf N}$. Erdős and Turán conjectured that for any basis $A$ of order 2 of ${\bf N}$, $\sigma_A(n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis $A$ of order 2 of ${\bf N}$ for which $\sigma_A(n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer $m_0$ such that, for any integer $m\geq m_0$, we have a set $A\subset {\bf Z}_m$ such that $A+A={\bf Z}_m$ and $\sigma_A(\overline{n})\leq 768$ for all $\overline{n}\in {\bf Z}_m$.

Détail

DOI : 10.4064/cm104-1-6

Mots-clés : sigma where subset erd tur conjectured basis order sigma unbounded imre nbsp ruzsa constructed basis order nbsp which sigma bounded square mean paper there exists positive integer integer geq have set subset sigma overline leq overline