Distribution of values of linear functions and asymptotic behavior of trajectories of some dynamical systems (Q1916636)

Let \(\xi_k=(\{\alpha_1k\}, \dots, \{\alpha_sk\})\), \(k\in\mathbb{N}\), where \(\alpha_1, \dots, \alpha_s \in\mathbb{R}\), which along with 1 are linearly independent over \(\mathbb{Z}\). Denote the deviation of the sequence \(\xi_k\), \(k=1, \dots,q\), by \[ D_q=\sup_{\gamma_1, \dots, \gamma_s \in[0,1)} |N_q(\gamma_1, \dots, \gamma_s)-\gamma_1 \dots \gamma_sq |, \] where \(N_q(\gamma_1, \dots, \gamma_s)\) is the number of \(\xi_k\) \((1\leq k\leq q)\) such that \(\{\alpha_jk\} < \gamma_j\), \(j=1, \dots, s\). The author constructs numbers \(\alpha_1, \dots, \alpha_s\) admitting perfect diophantine approximation, but the corresponding sequence \(\xi_k\), \(k=1, \dots, q\), is badly distributed in the unit square. One of these results is as follows: Let \(\varphi(y)\) be a positive monotone function such that \(\varphi(y) \downarrow 0\) arbitrarily slowly as \(y\to\infty\), and \(\psi(y)\) be a positive function monotone decreasing to zero as \(y\to\infty\) such that \(\varphi ((4\psi(y/8))^{-1}) < y^{-4}\) for all \(y\geq 1\). Then we can construct two numbers \(\alpha_1\) and \(\alpha_2\) such that the sequence \(\xi_k= (\{\alpha_1k\}, \{\alpha_2k\})\), \(k=1, \dots, q\), satisfies \(D_q \gg q\varphi (q)\) for all \(q\in\mathbb{N}\). Using these results, the author studies the asymptotic behavior of mean values of integrals of quasiperiodic functions, for example, to prove the nonrecurrence of mean values for arbitrarily smooth three-frequency quasiperiodic functions.