Distribution of values of linear functions and asymptotic behavior of trajectories of some dynamical systems

The asymptotic behavior of mean values for integrals of quasiperiodic functions, which characterizes the uniformity of the distribution of irrational windings on a torus, is shown to be essentially dependent on the dimension of the torus. We prove the nonrecurrence of mean values for arbitrarily smooth three-frequency quasiperiodic functions. We also present a series of results concerning the distribution of fractional parts for systems of linear functions.