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PreprintCC BY-NC-ND 4.0arXiv

On additive bases of finite abelian groups

  • Weidong Gao
  • Hanbin Zhang
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Abstract

Let G be a finite abelian group, let p be the smallest prime divisor of |G|, and let c0(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 c0(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 Cp⊕ Cpn turn out to be exceptional). In this paper, we continue the study of c0(G) and obtain the following results. (1) Together with known results, we prove that c0(G)=|G|/p+p−2 for all abelian groups G with rank(G)≥ 3. (2) For distinct odd primes p<q, we determine c0(Cp⊕ Cpq)=pq+2p−3, c0(Cq⊕ Cpq)=q2+p−2. (3) Let p be an odd prime, let 2≤ a≤ b, and let n be a positive integer with no prime divisor smaller than p. Then c0(Cpa⊕ Cpbn)=pa+b−1n+p−2. (4) For every odd integer n≥ 3, we prove c0(C3⊕ C3n)=3n+3. Therefore, c0(G) is determined except possibly for groups of the form Cp⊕ Cpn with p≥ 5.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026additive,
  title = {{On additive bases of finite abelian groups}},
  author = {Weidong Gao and Hanbin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37136},
  eprint = {2610.37136},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37136},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On additive bases of finite abelian groups
AU  - Weidong Gao
AU  - Hanbin Zhang
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37136
UR  - https://arxiv.org/abs/2610.37136
AB  - 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⊕ 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 rank(G)≥ 3. (2) For distinct odd primes p<q, we determine c_0(C_p⊕ C_pq)=pq+2p−3, c_0(C_q⊕ C_pq)=q^2+p−2. (3) Let p be an odd prime, let 2≤ a≤ b, and let n be a positive integer with no prime divisor smaller than p. Then c_0(C_p^a⊕ C_p^bn)=p^a+b−1n+p−2. (4) For every odd integer n≥ 3, we prove c_0(C_3⊕ C_3n)=3n+3. Therefore, c_0(G) is determined except possibly for groups of the form C_p⊕ C_pn with p≥ 5.
ER  -