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Constructions of high dimensional caps, sets without arithmetic progressions, and sets without zero sums Elsholtz, Christian
Description
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}
Item Metadata
Title |
Constructions of high dimensional caps, sets without arithmetic progressions, and sets without zero sums
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-11-14T09:01
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Description |
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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Extent |
43.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Graz University of Technology
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Series | |
Date Available |
2021-01-17
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0395635
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Researcher
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International