The problem to effectively construct uniformly distributed grids on a multidimensional unit cube is an actively developing and urgent direction with numerous applications and relations to other areas of mathematics (see, e.g., [1] and the bibliography therein). In the present paper, by a uniform distribution of a sequence of grids (finite sets) {ξ k } p k=1 in some s-dimensional unit cube [0, 1]s that are indexed by a sufficiently dense increasing sequence of positive integers p we mean the existence of positive numbers c(s) > 0 and β(s) > 0 such that the inequality