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Davenport constants with growing binary rank

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by Chao Liu ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We study the correction to the standard lower bound D∗D^* for Davenport constants of finite abelian groups with a growing elementary two-component. For every fixed nontrivial abelian pp-group HH of odd order, we prove D(H⊕C2r)=D∗(H⊕C2r)+(D(H)−1)log⁡2r+OH(1)D(H\oplus C_2^r)=D^*(H\oplus C_2^r)+(D(H)-1)\log_2 r+O_H(1), with a uniform version for varying HH. For C3⊕C2rC_3\oplus 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 C24⊕C2rC_{24}\oplus C_2^r, the ratio (D−D∗)/log⁡2r(D-D^*)/\log_2 r has limsup at least 2.4933231…>22.4933231\ldots>2, whereas for C3⊕C2rC_3\oplus C_2^r the limit is 22. These upper limiting coefficients are unbounded even with the odd prime fixed. The proofs use binary codes and finite-field Fourier analysis.

Provenance statement

OpenAI's ChatGPT was used to assist with mathematical exploration, proof development and checking, and manuscript drafting and revision.

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Version history

  1. v1Submitted by Chao LiuInitial depositCurrentOct 29, 2026