Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An extension of quantitative nondivergence and applications to Diophantine exponents


Author: Dmitry Kleinbock
Journal: Trans. Amer. Math. Soc. 360 (2008), 6497-6523
MSC (2000): Primary 37A17; Secondary 11J83
DOI: https://doi.org/10.1090/S0002-9947-08-04592-3
Published electronically: June 26, 2008
MathSciNet review: 2434296
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a sharpening of nondivergence estimates for unipotent (or more generally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approximation, it yields precise formulas for Diophantine exponents of affine subspaces of $ \mathbb{R}^{n}$ and their nondegenerate submanifolds.


References [Enhancements On Off] (What's this?)

  • [Be] V. Beresnevich, A Groshev type theorem for convergence on manifolds, Acta Math. Hungar. 94 (2002), 99-130. MR 1905790 (2003d:11109)
  • [BBDD] V. Beresnevich, V. Bernik, H. Dickinson, and M.M. Dodson, On linear manifolds for which the Khinchin approximation theorem holds, Vestsi Nats. Acad. Navuk Belarusi. Ser. Fiz.-Mat. Navuk (2000), 14-17 (Belorussian). MR 1820985 (2001j:11068)
  • [BBKM] V. Beresnevich, V. Bernik, D. Kleinbock, and G.A. Margulis, Metric Diophantine approximation: the Khintchine-Groshev theorem for non-degenerate manifolds, Moscow Math. J. 2 (2) (2002), 203-225. MR 1944505 (2004b:11107)
  • [BD] V. Bernik and M.M. Dodson, Metric Diophantine approximation on manifolds, Cambridge Univ. Press, Cambridge, 1999. MR 1727177 (2001h:11091)
  • [BKM] V. Bernik, D. Kleinbock, and G.A. Margulis, Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions, Internat. Math. Res. Notices (9) (2001), 453-486. MR 1829381 (2002g:11102)
  • [C] J.W.S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Math., vol. 45, Cambridge Univ. Press, Cambridge, 1957. MR 0087708 (19:396h)
  • [Da1] S.G. Dani, On invariant measures, minimal sets, and a lemma of Margulis, Invent. Math. (51) (1979), 239-260. MR 530631 (80d:58039)
  • [Da2] -, On orbits of unipotent flows on homogeneous spaces, Ergod. Th. Dynam. Sys. (4) (1984), 25-34. MR 758891 (86b:58068)
  • [Da3] -, Divergent trajectories of flows on s and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55-89. MR 794799 (87g:58110a)
  • [Da4] -, On orbits of unipotent flows on homogeneous spaces, II, Ergod. Th. Dynam. Sys. (6) (1986), 167-182. MR 857195 (88e:58052)
  • [Do] M.M. Dodson, Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation, J. Reine Angew. Math. 432 (1992), 69-76. MR 1184759 (94a:11125)
  • [G1] A. Ghosh, A Khintchine-type theorem for hyperplanes, J. London Math. Soc. 72 (2) (2005), 293-304. MR 2156655 (2006h:11089)
  • [G2] -, Metric Diophantine approximation over a local field of positive characteristic, J. Number Theory 124 (2) (2007), 454-469. MR 2321374 (2008g:11120)
  • [G3] -, Dynamics on homogeneous spaces and Diophantine approximation on manifolds, Ph.D. Thesis, Brandeis University, 2006.
  • [K1] D. Kleinbock, Some applications of homogeneous dynamics to number theory, in: Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), Proc. Symp. Pure Math., vol. 68, Amer. Math. Soc., Providence, RI, 2001, pp. 639-660. MR 1858548 (2002g:37009)
  • [K2] -, Extremal subspaces and their submanifolds, Geom. Funct. Anal. 13 (2) (2003), 437-466. MR 1982150 (2004f:11073)
  • [K3] -, Baker-Sprindžuk conjectures for complex analytic manifolds, in: Algebraic groups and Arithmetic, TIFR, India, 2004, pp. 539-553.
  • [KLW] D. Kleinbock, E. Lindenstrauss, and B. Weiss, On fractal measures and Diophantine approximation, Selecta Math. 10 (4) (2004), 479-523. MR 2134453 (2006g:11151)
  • [KM] D. Kleinbock and G.A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math. 148 (1998), 339-360. MR 1652916 (99j:11083)
  • [KT] D. Kleinbock and G. Tomanov, Flows on $ S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comm. Math. Helv. 82 (2007), 519-581. MR 2314053
  • [KW1] D. Kleinbock and B. Weiss, Badly approximable vectors on fractals, Israel J. Math. 149 (2005), 137-170. MR 2191212
  • [KW2] -, Friendly measures, homogeneous flows and singular vectors, in: Algebraic and Topological Dynamics, Contemp. Math., vol. 385, AMS, Providence, RI, 2005, pp. 281-292. MR 2180240 (2006f:11084)
  • [KW3] -, Dirichlet's theorem on diophantine approximation and homogeneous flows, J. Mod. Dyn. 2 (1) (2008), 43-62. MR 2366229
  • [Mr1] G.A. Margulis, On the action of unipotent group in the space of lattices, Proceedings of the Summer School on group representations (Budapest 1971), Académiai Kiado, Budapest, 1975, pp. 365-370. MR 0470140 (57:9907)
  • [Mr2] -, Diophantine approximation, lattices and flows on homogeneous spaces, in: A panorama of number theory or the view from Baker's garden (Zürich, 1999), Cambridge Univ. Press, Cambridge, 2002, pp. 280-310. MR 1975458 (2004h:11031)
  • [Mt] P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995. MR 1333890 (96h:28006)
  • [PV] A. Pollington and S. Velani, Metric Diophantine approximation and `absolutely friendly' measures, Selecta Math. 11 (2) (2005), 297-307. MR 2183850 (2006k:11142)
  • [Rg] M.S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, Berlin and New York, 1972. MR 0507234 (58:22394a)
  • [Rt1] M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235-280. MR 1106945 (93f:22012)
  • [Rt2] -, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994), 236-257. MR 1262705 (95c:22018)
  • [Sc1] W. Schmidt, Diophantine approximation and certain sequences of lattices, Acta Arith. 18 (1971), 178-195. MR 0286751 (44:3960)
  • [Sc2] -, Diophantine approximation, Springer-Verlag, Berlin and New York, 1980. MR 0568710 (81j:10038)
  • [Sp] V. Sprindžuk, Achievements and problems in Diophantine approximation theory, Russian Math. Surveys 35 (1980), 1-80. MR 586190 (81j:10039)
  • [SU] B. Stratmann and M. Urbański, Diophantine extremality of the Patterson measure, Math. Proc. Cambridge Phil. Soc. 140 (2006), no. 2, 297-304. MR 2212281 (2007g:11090)
  • [U] M. Urbański, Diophantine approximation of self-conformal measures, J. Number Th. 110 (2005), 219-235. MR 2122607 (2006a:11100)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37A17, 11J83

Retrieve articles in all journals with MSC (2000): 37A17, 11J83


Additional Information

Dmitry Kleinbock
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110
Email: kleinboc@brandeis.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04592-3
Received by editor(s): December 15, 2006
Published electronically: June 26, 2008
Additional Notes: This work was supported in part by NSF Grant DMS-0239463.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society