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A note on minimal dispersion of point sets in the unit cube. (arXiv:1707.08794v2 [cs.CG] UPDATED)
来源于:arXiv
We study the dispersion of a point set, a notion closely related to the
discrepancy. Given a real $r\in (0,1)$ and an integer $d\geq 2$, let $N(r,d)$
denote the minimum number of points inside the $d$-dimensional unit cube
$[0,1]^d$ such that they intersect every axis-aligned box inside $[0,1]^d$ of
volume greater than $r$. We prove an upper bound on $N(r,d)$, matching a lower
bound of Aistleitner et al. up to a multiplicative constant depending only on
$r$. This fully determines the rate of growth of $N(r,d)$ if $r\in(0,1)$ is
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