Deterministic constructions of high-dimensional sets with small dispersion

Mario Ullrich; Jan Vybíral

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arXiv · 1901.06702

Deterministic constructions of high-dimensional sets with small dispersion

Mario Ullrich·Jan Vybíral

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Abstract

The dispersion of a point set $P\subset[0,1]^d$ is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect $P$. Here, we show a construction of low-dispersion point sets, which can be deduced from solutions of certain $k$-restriction problems, which are well-known in coding theory. It was observed only recently that, for any $\varepsilon>0$, certain randomized constructions provide point sets with dispersion smaller than $\varepsilon$ and number of elements growing only logarithmically in $d$. Based on deep results from coding theory, we present explicit, deterministic algorithms to construct such point sets in time that is only polynomial in $d$. Note that, however, the running-time will be super-exponential in $\varepsilon^{-1}$.

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Mario Ullrich, Jan Vybíral. 2019-01-20. Deterministic constructions of high-dimensional sets with small dispersion. https://arxiv.org/abs/1901.06702

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