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 Chebyshevskii Sb., 2019, Volume 20, Issue 2, Pages 207–220 (Mi cheb764)

Trigonometric sums in the metric theory of Diophantine approximation

E. I. Kavaleuskaya

Belarusian State Agricultural Technic University (Minsk)

Abstract: 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 $\Gamma$, $\dim\Gamma=m$, $n/2<m<n$. 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.

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

DOI: https://doi.org/10.22405/2226-8383-2018-20-2-207-220

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UDC: 511.36
Accepted:12.07.2019

Citation: E. I. Kavaleuskaya, “Trigonometric sums in the metric theory of Diophantine approximation”, Chebyshevskii Sb., 20:2 (2019), 207–220

Citation in format AMSBIB
\Bibitem{Kov19} \by E.~I.~Kavaleuskaya \paper Trigonometric sums in the metric theory of Diophantine approximation \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 2 \pages 207--220 \mathnet{http://mi.mathnet.ru/cheb764} \crossref{https://doi.org/10.22405/2226-8383-2018-20-2-207-220}