2021/07/22 by Kurt S. MacKay, MacKay, Kurt S.
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Limits and Structures in Graph Theory #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2107.10453
openalex publication_date 2021/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. Let d>1. The dispersion of T ⊂ [0,1]d is defined as the supremum of the volume taken over all axis parallel boxes in the cube which do not intersect T. The minimal dispersion of n points in the cube is defined as the infimum of the dispersion taken over all T such that |T| = n. Define the "large volume" regime as the class of all volumes (1)/(4) < r ≤ (1)/(2). The inverse of the minimal dispersion is denoted as N(r,d). When the volume is large, the best known upper bound on N(r,d) is of the order (r - (1)/(4))-1. The construction presented in this note yields an upper bound given by N(r,d) ≤ \lfloor \fracπ√(r - (1)/(4)) \rfloor - 3 . Some of our intermediate estimates are sharp given the condition that d ≥ Cr, where Cr is a positive constant which depends only on the volume r.