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On additive bases III

Volume 193 / 2020

Weidong Gao, Yongke Qu, Hanbin Zhang Acta Arithmetica 193 (2020), 293-308 MSC: Primary 11B75; Secondary 11P70, 20K01. DOI: 10.4064/aa181106-22-4 Published online: 24 January 2020

Abstract

Let $G$ be an additive finite abelian group and $S$ a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H\subset G$, $S$ 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 $|S|\ge t$ forms an additive basis of $G$, i.e., every element of $G$ can be expressed as a sum over a nonempty subsequence of $S$. Some related problems on $\mathsf c_0(G)$ have been studied by Mann and Olson around 1968. Answering a problem proposed by Olson, Peng determined $\mathsf c_0(G)$ for all elementary finite abelian groups in 1987. Recently, several authors have studied $\mathsf c_0(G)$ for general abelian groups, and so far the invariant $\mathsf c_0(G)$ has been determined for some finite abelian groups including the cyclic groups, the groups of even order, the groups of rank at least five, the groups of rank in $\{3, 4\}$ and with the smallest prime divisor of $|G|$ at least 11, and the $p$-groups of rank 2. In this paper, we shall determine $\mathsf c_0(G)$ for all groups such that the rank of $G$ is in $\{3, 4\}$, the smallest prime divisor of $|G|$ does not exceed 7 and $|G|\ge 3.72\times 10^7$.

Authors

  • Weidong GaoCenter for Combinatorics, LPMC
    Nankai University
    300071 Tianjin, P.R. China
    e-mail
  • Yongke QuDepartment of Mathematics
    Luoyang Normal University
    471022 Luoyang, P.R. China
    e-mail
  • Hanbin ZhangAcademy of Mathematics and Systems Science
    Chinese Academy of Sciences
    100190 Beijing, P.R. China
    e-mail

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