A basis of $ \mathbb{Z}_m$, II

Min Tang¹ ; Yong-Gao Yong-Gao²

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

Colloquium Mathematicum, Tome 108 (2007) no. 1, pp. 141-145

Résumé

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 $\overline {n}\in \mathbb{Z}_m$.

Détail

DOI : 10.4064/cm108-1-12

Keywords: given set subset mathbb sigma denote number ordered pairs times erd tur conjectured asymptotic basis mathbb sigma unbounded analogue erd tur conjecture does abelian group mathbb namely natural number there exists set subseteq mathbb mathbb sigma overline leq overline mathbb