Trigonometric sums in the metric theory of Diophantine approximation

E. I. Kavaleuskaya

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 207-220

Résumé

It is a survey with respect to using trigonometric sums in the metric theory of Diophantine approximation on the manifolds in $n$-dimensional Euclidean space. We represent both classical results and contemporary theorems for $\Gamma$, $\dim\Gamma=m$, $n/2$. We also discuss reduction of a problem about Diophantine approximation to trigonometric sum or trigonometric integral, and indicate measure-theoretic considerations. If $m\le n/2$ then usually it is used the other methods. For example, the essential and inessential domains method or methods of Ergodic Theory. Here we cite two fundamental theorems of this theory. One of them was obtained by V. G. Sprindzuk (1977). The other theorem was proved by D. Y. Kleinbock and G. A. Margulis (1998). The first result was obtained using method of trigonometric sums. The second theorem was proved using methods of Ergodic Theory. Here the authors applied new technique which linked Diophantine approximation and homogeneous dynamics. In conclusion, we add a short comment concerning the tendencies of a development of the metric theory of Diophantine approximation of dependent quantities and its contemporary aspects.

Détail

Keywords: Diophantine approximation, metric theory, differentiable manifolds, trigonometric sums, Van der Corput's method, I. M. Vinogradov's method of trigonometric sums.