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An asymptotically tight bound for the Davenport constant

Banff International Research Station for Mathematical Innovation and Discovery

In this talk, we will present a new upper bound for the Davenport constant of finite Abelian groups of the form $C^r_n$. An old conjecture in zero-sum theory asserts that $\mathsf{D}(C^r_n) = r(n-1) + 1$ holds for all integers $n,r \geqslant 1$ and still widely stands to this day. In this context, our bound turns out to be particularly relevant as it implies that for every integer $r \geqslant 1$, the Davenport constant $\mathsf{D}(C^r_n)$ is asymptotic to $rn$ when $n$ tends to infinity, thus proving the conjecture in an asymptotic sense. This improves on the best previously known upper bound which was $\mathsf{D}(C^r_n) \leqslant n(1 + (r-1)\log n)$. An extension of our theorem to a wider framework as well as related open problems will also be discussed.

Mathematics; Combinatorics; Number theory; Discrete mathematics

video/mp4

Author affiliation: Sorbonne Université

2020-05-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74386

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/