We give a new lower bound for the minimal dispersion of a point set in the unit cube and its inverse function in the high dimension regime. This is done by considering only a very small class of test boxes, which allows us to reduce bounding the dispersion to a problem in extremal set theory. Specifically, we translate a lower bound on the size of $r$-cover-free families to a lower bound on the inverse of the minimal dispersion of a point set. The lower bound we obtain matches the recently obtained upper bound on the minimal dispersion up to logarithmic terms.

翻译：我们给出了高维情形下单位立方体点集最小分散度及其逆函数的一个新下界。通过仅考虑一类非常小的测试盒子，我们将分散度的界定问题转化为极值集合论中的一个问题。具体而言，我们将$r$-无覆盖族规模的下界转化为点集最小分散度逆函数的下界。所得下界与最近获得的最小分散度上界在对数项上达到匹配。