{
  "abstract": "Let $G$ be a finite abelian group, let $p$ be the smallest prime divisor of $|G|$, and let $c_0(G)$ be the least integer $t$ such that every regular sequence over $G$ of length at least $t$ is an additive basis. Additive bases have been extensively studied in additive combinatorics, and the exact value of $c_0(G)$ has been determined for many groups of rank at least three and some groups of rank two (in which case groups of the form $C_p\\oplus C_{pn}$ turn out to be exceptional). In this paper, we continue the study of $c_0(G)$ and obtain the following results. (1) Together with known results, we prove that $c_0(G)=|G|/p+p-2$ for all abelian groups $G$ with $\\text{rank}(G)\\geq 3$. (2) For distinct odd primes $p<q$, we determine \\[ c_0(C_p\\oplus C_{pq})=pq+2p-3,\\qquad c_0(C_q\\oplus C_{pq})=q^2+p-2. \\] (3) Let $p$ be an odd prime, let $2\\leq a\\leq b$, and let $n$ be a positive integer with no prime divisor smaller than $p$. Then \\[ c_0(C_{p^a}\\oplus C_{p^bn})=p^{a+b-1}n+p-2. \\] (4) For every odd integer $n\\geq 3$, we prove \\[ c_0(C_3\\oplus C_{3n})=3n+3. \\] Therefore, $c_0(G)$ is determined except possibly for groups of the form $C_p\\oplus C_{pn}$ with $p\\geq 5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37136",
  "author_total": 2,
  "authors": [
    {
      "name": "Weidong Gao"
    },
    {
      "name": "Hanbin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1nsmmps9k4v78.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37136",
  "fulltext_url": null,
  "id": "1nsmmps9k4v78",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37136",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7224091287",
  "pdf_url": "https://arxiv.org/pdf/2610.37136",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1nsmmps9k4v78.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37136",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37136"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On additive bases of finite abelian groups",
  "title_text": "On additive bases of finite abelian groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/1nsmmps9k4v78",
  "venue": "arXiv",
  "versions": []
}