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On additive bases II

Volume 168 / 2015

Weidong Gao, Dongchun Han, Guoyou Qian, Yongke Qu, Hanbin Zhang Acta Arithmetica 168 (2015), 247-267 MSC: 11P70, 11B50, 11B75. DOI: 10.4064/aa168-3-3

Abstract

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 $n\geq 2.$

Authors

  • Weidong GaoCenter for Combinatorics
    Nankai University
    Tianjin 300071, P.R. China
    e-mail
  • Dongchun HanCenter for Combinatorics
    Nankai University
    Tianjin 300071, P.R. China
    e-mail
  • Guoyou QianMathematical College
    Sichuan University
    Chengdu 610064, P.R. China
    e-mail
  • Yongke QuDepartment of Mathematics
    Luoyang Normal University
    Luoyang 471022, P.R. China
    e-mail
  • Hanbin ZhangCenter for Combinatorics
    Nankai University
    Tianjin 300071, P.R. China
    e-mail

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