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Constructions of high dimensional caps, sets without arithmetic progressions, and sets without zero sums

Banff International Research Station for Mathematical Innovation and Discovery

In this talk we discuss the following problems. \begin{enumerate} \item For a finite abelian group $G$ let $\mathsf s (G)$ denote the smallest integer $l$ such that every sequence $S$ over $G$ of length $|S| \ge l$ has a zero-sum subsequence of length $\exp (G)$. Specialising to $G=\mathbb{Z}_n^r$, the Erd\H{o}s-Ginzburg-Ziv theorem states that $\mathsf s (\mathbb{Z}_n)=2n-1$ and Reiher proved that $\mathsf s (\mathbb{Z}_n^2)=4n-3$. The speaker proved (some years ago) that for odd $n$ $\mathsf s (\mathbb{Z}_n^3)\geq 9n-8$ and (jointly with Edel, Geroldinger, Kubertin and Rackham) $\mathsf s (\mathbb{Z}_n^4)\geq 20n-19$. It is an open problem if possibly $\mathsf s (\mathbb{Z}_n^3)=9n-8$ holds for odd $n$, or not. \item Let $r_k(\mathbb{Z}_m^r)$ denote the maximal size of a set in $\mathbb{Z}_m^r$ without an arithmetic progression of $k$ distinct elements. There has been major progress in recent years. Croot, Lev and Pach considerably improved the upper bound of $r_3(\mathbb{Z}_4^r)$, and Ellenberg and Gijswijt extended this to $r_3(\mathbb{F}_q^r)$. This is the famous cap set problem. In joint work with Pach we considerably improved the constructions for lower bounds of $r_k(\mathbb{Z}_m^r)$. In particular $r_3(\mathbb{Z}_4^r)\geq \frac{9}{4\sqrt{\pi}} \cdot \frac{3^r}{\sqrt{r}}$. \end{enumerate}

Mathematics; Combinatorics; Number theory; Discrete mathematics

video/mp4

Author affiliation: Graz University of Technology

2021-01-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77125

Unreviewed

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