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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 Khinchin and Jarnik theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using 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.
|Number of pages||22|
|Early online date||20 Jun 2014|
|Publication status||Published - Aug 2014|
- automorphic spectrum
- Diophantine approximation
- homogeneous varieties
- Jarník theorem
- Khintchine theorem
- semisimple algebraic groups
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- 1 Finished
Dynamics of large Group Actions, Rigidity and Diophantine Geometry
1/02/10 → 1/02/13