On additive bases II

Weidong Gao ¹   ; Dongchun Han ¹   ; Guoyou Qian ²   ; Yongke Qu ³   ; Hanbin Zhang ¹

¹ Center for Combinatorics Nankai University Tianjin 300071, P.R. China
² Mathematical College Sichuan University Chengdu 610064, P.R. China
³ Department of Mathematics Luoyang Normal University Luoyang 471022, P.R. China

Acta Arithmetica, Tome 168 (2015) no. 3, pp. 247-267

Résumé

Let $G$ be an additive finite abelian group, and let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subseteq G$, $S$ contains at most $|H|-1$ terms from $H$. Let $\mathsf c_0(G)$ be the smallest integer $t$ such that every regular sequence $S$ over $G$ of length $|S|\geq t$ forms an additive basis of $G$, i.e., every element of $G$ can be expressed as the sum over a nonempty subsequence of $S$. The constant $\mathsf c_0(G)$ has been determined previously only for the elementary abelian groups. In this paper, we determine $\mathsf c_0(G)$ for some groups including the cyclic groups, the groups of even order, the groups of rank at least five, and all the $p$-groups except $G=C_p\oplus C_{p^n}$ with $n\geq 2.$

Détail

DOI : 10.4064/aa168-3-3

Keywords: additive finite abelian group sequence say regular every proper subgroup subseteq contains terms mathsf smallest integer every regular sequence length geq forms additive basis every element expressed sum nonempty subsequence constant mathsf has determined previously only elementary abelian groups paper determine mathsf groups including cyclic groups groups even order groups rank least five p groups except oplus geq