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Davenport constants with growing binary rank

  • Chao Liu
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Abstract

We study the correction to the standard lower bound D* for Davenport constants of finite abelian groups with a growing elementary two-component. For every fixed nontrivial abelian p-group H of odd order, we prove D(H⊕ C2r)=D*(H⊕ C2r)+(D(H)−1)log2 r+OH(1), with a uniform version for varying H. For C3⊕ C2r, we determine exact values on explicit infinite intervals of ranks. More generally, the correction is logarithmic or smaller precisely when the fixed group has at most one odd prime divisor. We show that the two-primary part affects the logarithmic coefficient: for C24⊕ C2r, the ratio (D−D*)/log2 r has limsup at least 2.4933231…>2, whereas for C3⊕ C2r the limit is $2$. These upper limiting coefficients are unbounded even with the odd prime fixed. The proofs use binary codes and finite-field Fourier analysis.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026davenport,
  title = {{Davenport constants with growing binary rank}},
  author = {Chao Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38447},
  eprint = {2610.38447},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38447},
}
RIS
TY  - UNPB
TI  - Davenport constants with growing binary rank
AU  - Chao Liu
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38447
UR  - https://arxiv.org/abs/2610.38447
AB  - We study the correction to the standard lower bound D^* for Davenport constants of finite abelian groups with a growing elementary two-component. For every fixed nontrivial abelian p-group H of odd order, we prove D(H⊕ C_2^r)=D^*(H⊕ C_2^r)+(D(H)−1)log_2 r+O_H(1), with a uniform version for varying H. For C_3⊕ C_2^r, we determine exact values on explicit infinite intervals of ranks. More generally, the correction is logarithmic or smaller precisely when the fixed group has at most one odd prime divisor. We show that the two-primary part affects the logarithmic coefficient: for C_24⊕ C_2^r, the ratio (D−D^*)/log_2 r has limsup at least 2.4933231…>2, whereas for C_3⊕ C_2^r the limit is $2$. These upper limiting coefficients are unbounded even with the odd prime fixed. The proofs use binary codes and finite-field Fourier analysis.
ER  -