A note on minimal dispersion of point sets in the unit cube. (March 2018)

• Record Type: Journal Article
• Authors: Sosnovec, Jakub
• Is Part Of: European journal of combinatorics. Volume 69(2018)
• Page Start: 255
• Page End: 259
• Publication Date: 2018-03
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2017.11.006
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real r ∈ ( 0, 1 ) and an integer d ≥ 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 ∈ ( 0, 1 ) is fixed.