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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 [email protected] 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 [email protected]
Amos Nevo
Affiliation:
Department of Mathematics, Technion IIT, Israel email [email protected]
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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 

References

Beresnevich, V. and Velani, S., A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164 (2006), 971–992.Google Scholar
Bernik, V. and Dodson, M. M., Metric diophantine approximation on manifolds, Cambridge Tracts in Mathematics, vol. 137 (Cambridge University Press, Cambridge, 1999).Google Scholar
Blomer, V. and Brumley, F., On the Ramanujan conjecture over number fields, Ann. of Math. (2) 174 (2011), 581–605.Google Scholar
Clozel, L., Automorphic forms and the distribution of points on odd-dimensional spheres, Israel J. Math. 132 (2002), 175–187.CrossRefGoogle Scholar
Clozel, L., Démonstration de la conjecture τ, Invent. Math. 151 (2003), 297–328.Google Scholar
Dani, S. G., Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55–89.Google Scholar
Drutu, C., Diophantine approximation on rational quadrics, Math. Ann. 333 (2005), 405–469.Google Scholar
Ghosh, A., Gorodnik, A. and Nevo, A., Diophantine approximation and automorphic spectrum, Int. Math. Res. Not. IMRN 2012; doi:10.1093/imrn/rns198.CrossRefGoogle Scholar
Ghosh, A., Gorodnik, A. and Nevo, A., Diophantine approximation exponents on homogeneous varieties, Contemp. Math., to appear, arXiv:1401.6581 [math.NT].Google Scholar
Gorodnik, A. and Nevo, A., Quantitative ergodic theorems and their number-theoretic applications, Preprint.Google Scholar
Harman, G., Metric number theory, London Mathematical Society Monographs, vol. 18 (Clarendon Press, New York, 1998).Google Scholar
Jarník, V., Diophantische Approximationen und Hausdorffsches Mass, Mat. Sb. 36 (1929), 371–382.Google Scholar
Khintchine, A., Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), 115–125.Google Scholar
Kim, H., Functoriality for the exterior square of GL4and the symmetric fourth of GL2. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak, J. Amer. Math. Soc. 16 (2003), 139–183.Google Scholar
Kleinbock, D., Quantitative nondivergence and its Diophantine applications, in Homogeneous flows, moduli spaces and arithmetic, Clay Mathematics Proceedings, vol. 10 (American Mathematical Society, Providence, RI, 2010), 131–153.Google Scholar
Kleinbock, D. and Margulis, G., Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), 451–494.CrossRefGoogle Scholar
Kleinbock, D. and Merrill, K., Rational approximation on spheres, Preprint (2013),arXiv:1301.0989.Google Scholar
Lang, S., Report on Diophantine approximation, Bull. Soc. Math. France 93 (1965), 177–192.CrossRefGoogle Scholar
Lubotzky, A., Discrete groups, expanding graphs and invariant measures, with an appendix by Jonathan D. Rogawski, Progress in Mathematics, vol. 125 (Birkhäuser Verlag, Basel, 1994).Google Scholar
Platonov, V. and Rapinchuk, A., Algebraic groups and number theory (Academic Press, New York, 1994).Google Scholar
Sarnak, P., Notes on the generalized Ramanujan conjectures, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 659–685.Google Scholar
Shin, S. W., Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), 1645–1741.Google Scholar
Sprindzuk, V. G., Metric theory of Diophantine approximations, Scripta Series in Mathematics (V. H. Winston, Washington, DC, 1979).Google Scholar