
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 measuretheoretic
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/10.22405/222683832018202207220
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UDC:
511.36 Received: 14.05.2019 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 207220
\mathnet{http://mi.mathnet.ru/cheb764}
\crossref{https://doi.org/10.22405/222683832018202207220}
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