Hexagon

math.NT — Number Theory

Davenport constants with growing binary rank

Contributed by Chao Liu

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 $D(H\oplus C_2^r)=D^*(H\oplus C_2^r)+(D(H)-1)\log_2 r+O_H(1)$ , with a uniform version for varying $H$ . For $C_3\oplus C_2^r$ , 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 $C_{24}\oplus C_2^r$ , the ratio $(D-D^*)/\log_2 r$ has limsup at least $2.4933231\ldots>2$ , whereas for $C_3\oplus C_2^r$ 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.

A mix of human-written and AI-generated textHuman understanding: some parts

Oct 29, 2026PDF ↗View work →

hexagon:2610.00599v1

math.NT — Number Theory

A logarithmic error term for subset sums of zero-sum-free sets

Contributed by Chao Liu

For a positive integer $k$ , let $f(k)$ be the minimum number of distinct nonempty subset sums of a zero-sum-free $k$ -element set, where the minimum is taken over all finite abelian groups. We prove $f(k)\ge \frac{k^2}{2}-13k\log(k+1)-116k$ . Together with the classical upper bound, this gives $f(k)=k^2/2+O(k\log k)$ , with the optimal leading coefficient. As a consequence, the Olson constant of every finite abelian group $G$ satisfies $\mathrm{Ol}(G)\le\sqrt{2|G|}+O(\log|G|)$ . The argument combines a triangular lower bound for repeated supports, a count using blocks of two and three copies, and a greedy expansion argument. No restriction is imposed on the order of the group or on the stabilizer of the set of subset sums.

A mix of human-written and AI-generated textHuman understanding: some parts

Oct 29, 2026PDF ↗View work →

Submitted works

Davenport constants with growing binary rank

A logarithmic error term for subset sums of zero-sum-free sets