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Tuesday, November 28, 2017
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11:00am [11:00am]Anand Srivastav Kiel University Germany
Description:
One-Sided Multicolor Discrepancy of Hyperplanes over Finite Fields Anand Srivastav Kiel University Germany Abstract: We investigate the multicolor discrepancy and the one-sided multicolor discrepancy of linear hyperplanes in the finite vector space $F_{q}^{r}$. We show that the one-sided discrepancy is bounded from below by $\Omega_{q}\left(\sqrt{n/c}\right)$, $c$ the number of colors, using Fourier analysis on $\mathbb{F}_{q}^{r}$. We also show an upper bound of of $O_{q}(\sqrt{n\log c})$. The upper bound is derived by the $c$--color extension of Spencer's six standard deviation theorem and is also valid for the one-sided discrepancy. Thus, the gap between the upper and lower bound for the one-sided discrepancy is a factor of $\sqrt{c\log c}$ and the bounds are tight for any constant $c$ and $q$. For large $c$, more precisely for $c\geq qn^{1/3}$, we reduce this gap to a factor of $\sqrt{\log c}$. All together this exhibits a new example of a hypergraph with (almost) sharp discrepancy bounds.

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