A basis of $ \mathbb{Z}_m$, II
Volume 108 / 2007
Colloquium Mathematicum 108 (2007), 141-145
MSC: 11B13, 11B34.
DOI: 10.4064/cm108-1-12
Abstract
Given a set $A\subset \mathbb{N}$ let $\sigma_A(n)$ denote the number of ordered pairs $(a,a')\in A\times A$ such that $a+a'=n$. Erdős and Turán conjectured that for any asymptotic basis $A$ of $\mathbb{N}$, $\sigma_A(n)$ is unbounded. We show that the analogue of the Erdős–Turán conjecture does not hold in the abelian group $(\mathbb{Z}_m,+)$, namely, for any natural number $m$, there exists a set $A\subseteq\mathbb{Z}_m$ such that $A+A=\mathbb{Z}_m$ and $\sigma_A(\overline {n})\leq 5120$ for all $\overline {n}\in \mathbb{Z}_m$.
