A sharp discrepancy bound for jittered sampling

For m , d ? N m, \in {\mathbb N} , a jittered (or stratified) sampling point set alttext="upper P"> P encoding="application/x-tex">P having N equals m Superscript d"> N = encoding="application/x-tex">N = m^d points in alttext="left-bracket 0 1 right-parenthesis [ 0 1 stretchy="false">) encoding="application/x-tex">[0,1)^d is constructed by partitioning the unit cube into encoding="application/x-tex">m^d axis-aligned cubes of equal size and then placing one independently uniformly at random each cube. We show that there are constants alttext="c greater-than 0"> c > encoding="application/x-tex">c > 0 C"> C encoding="application/x-tex">C such for all alttext="d"> encoding="application/x-tex">d greater-than-or-equal-to ? maxsize="1.2em" minsize="1.2em">( / minsize="1.2em">) encoding="application/x-tex">\Theta \big (\big (\frac {1+\log (m/d)}{m/d}\big )^{1/2}\big ) than distributed (Monte Carlo set) cardinality . result improves both lower bound given Pausinger Steinerberger [J. Complexity 33 (2016), pp. 199–216]. It also removes asymptotic requirement alttext="m"> encoding="application/x-tex">m sufficiently large compared to