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Metric Diophantine approximation on homogeneous varieties

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Ergodic theory

Published online by Cambridge University Press:  20 June 2014

Anish Ghosh ,
Alexander Gorodnik  and
Amos Nevo
Anish Ghosh
Affiliation:
School of Mathematics, University of East Anglia, Norwich, UK email ghosh@math.tifr.res.in Current address: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India
Alexander Gorodnik
Affiliation:
School of Mathematics, University of Bristol, Bristol, UK email a.gorodnik@bristol.ac.uk
Amos Nevo
Affiliation:
Department of Mathematics, Technion IIT, Israel email anevo@tx.technion.ac.il
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Abstract

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We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khintchine and Jarník theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$-algebraic integers. In several important examples, the metric results we obtain are optimal. The proof uses quantitative equidistribution properties of suitable averaging operators, which are derived from spectral bounds in automorphic representations.

Keywords

Diophantine approximationKhintchine theoremJarník theoremsemisimple algebraic groupshomogeneous varietiesautomorphic spectrum

MSC classification

Secondary: 37A17: Homogeneous flows 11K60: Diophantine approximation
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 8 , August 2014 , pp. 1435 - 1456
Copyright
© The Author(s) 2014 

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