Multidimensional generalization of sums of fraction parts and their applications to number theory

In the paper a new multidimensional generalization of fraction part function is introduced. We obtain a formula which express the number of points from the orbit of irrational shift on multidimensional torus, lying in a given domain, in the terms of sums of multidimensional fraction parts. Also we give some application of this formula to various number-theoretic problems.