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Additive bases of $C_3\oplus C_{3q}$

Volume 169 / 2022

Yongke Qu, Yuanlin Li Colloquium Mathematicum 169 (2022), 189-195 MSC: Primary 11B75; Secondary 11P70. DOI: 10.4064/cm8515-6-2021 Published online: 15 February 2022


Let $G$ be a finite abelian group and $p$ be the smallest prime dividing $|G|$. Let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subsetneq G$, $S$ 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., $\sum (S)=G$. The invariant $\mathsf c_0(G)$ was first studied by Olson and Peng in 1980’s, and since then it has been determined for all finite abelian groups except for the groups with rank 2 and a few groups of rank 3 or 4 with order less than $10^8$. In this paper, we focus on the remaining case concerning groups of rank 2. It was conjectured by the first author and Han [Int. J. Number Theory 13 (2017), 2453–2459] that $\mathsf c_0(G)=pn+2p-3$ where $G=C_p\oplus C_{pn}$ with $n\geq 3$. We confirm the conjecture for the case when $p=3$ and $n=q$ $(\geq 5)$ is a prime number.


  • Yongke QuDepartment of Mathematics
    Luoyang Normal University
    Luoyang 471934, P.R. China
  • Yuanlin LiDepartment of Mathematics and Statistics
    Brock University
    St. Catharines, ON L2S 3A1, Canada

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