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BIRS Workshop Lecture Videos

An asymptotically tight bound for the Davenport constant Girard, Benjamin


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.

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