Minimal dispersion of large volume boxes in the cube

In this note we present a construction which improves the best known bound on minimal dispersion of large volume boxes in unit cube. Let d>1. The T⊂[0,1]d is defined as supremum taken over all axis parallel cube do not intersect T. n points infimum T such that |T|=n. Define “large volume” regime class volumes 14<r≤12. inverse denoted N(r,d). When large, upper N(r,d) order (r−14)−1. presented yields an given byN(r,d)≤⌊πr−14⌋−3. Some our intermediate estimates are sharp condition d≥Cr, where Cr positive constant depends only r.