Deterministic Constructions of High-Dimensional Sets with Small Dispersion

Abstract

AbstractThe dispersion of a point set $$P\subset [0,1]^d$$ P ⊂ [ 0 , 1 ] d is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect P. It was observed only recently that, for any $$\varepsilon >0$$ ε > 0 , certain randomized constructions provide point sets with dispersion smaller than $$\varepsilon $$ ε 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 $$\varepsilon ^{-1}$$ ε - 1 . Our construction is based on the apparently new insight that low-dispersion point sets can be deduced from solutions of certain k-restriction problems, which are well-known in coding theory.