A basis of ${\bf Z}_m$
Volume 104 / 2006
Colloquium Mathematicum 104 (2006), 99-103
MSC: 11B13, 11B34.
DOI: 10.4064/cm104-1-6
Abstract
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 $\overline{n}\in {\bf Z}_m$.
