Constructions of uniform distributions in terms of geometry of numbers

In the paper the author proves that the points of admissible lattices in the Euclidean space are distributed very uniformly in parallelepipeds. In particular, the remainder terms in the corresponding lattice point problem are found to be logarithmically small. As an application of these results point sets with the lowest possible discrepancies in the unit cube and quadrature formulas with the smallest possible errors in the classes of functions with anisotropic smoothness are given in terms of admissible lattices.