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


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}

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