Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms


Author: Nimish A. Shah
Journal: J. Amer. Math. Soc. 23 (2010), 563-589
MSC (2010): Primary 22E40, 11J83
Published electronically: December 29, 2009
MathSciNet review: 2601043
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski cannot be improved for almost all points on any analytic curve in $ \mathbb{R}^k$ which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late 1960s.

The Diophantine problem is then settled via showing that a certain sequence of expanding translates of curves in the homogeneous space of unimodular lattices in $ \mathbb{R}^{k+1}$ gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization techniques, and a new observation about intertwined linear dynamics of various $ {SL}(m,\mathbb{R})$'s containeod in $ {SL}(k+1,\mathbb{R})$.


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

  • 1. R. C. Baker.
    Dirichlet's theorem on Diophantine approximation.
    Math. Proc. Cambridge Philos. Soc., 83(1):37-59, 1978. MR 0485713 (58:5535)
  • 2. Y. Bugeaud.
    Approximation by algebraic integers and Hausdorff dimension.
    J. London Math. Soc. (2), 65(3):547-559, 2002. MR 1895732 (2003d:11110)
  • 3. J. W. S. Cassels.
    An introduction to the geometry of numbers.
    Die Grundlehren der mathematischen Wissenschaften, Band 99. Springer-Verlag, Berlin-New York, 1971. viii+344 pp. MR 0306130 (46:5257)
  • 4. S. G. Dani.
    Divergent trajectories of flows on homogeneous spaces and Diophantine approximation.
    J. Reine Angew. Math., 359:55-89, 1985. MR 794799 (87g:58110a)
  • 5. S. G. Dani and G. A. Margulis.
    Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces.
    Proc. Indian Acad. Sci. Math. Sci., 101(1):1-17, 1991. MR 1101994 (92g:22027)
  • 6. S. G. Dani and G. A. Margulis.
    Limit distributions of orbits of unipotent flows and values of quadratic forms.
    In I. M. Gelfand Seminar, pages 91-137. Amer. Math. Soc., Providence, RI, 1993. MR 1237827 (95b:22024)
  • 7. H. Davenport and W. M. Schmidt
    Dirichlet's theorem on diophantine approximation. II.
    Acta Arith., 16:413-424, 1969/1970. MR 0279040 (43:4766)
  • 8. H. Davenport and W. M. Schmidt.
    Dirichlet's theorem on diophantine approximation.
    Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pp. 113-132, Academic Press, London, 1970. MR 0272722 (42:7603)
  • 9. M. M. Dodson, B. P. Rynne, J. A. Vickers.
    Dirichlet's theorem and Diophantine approximation on manifolds.
    J. Number Theory, 36(1):85-88, 1990. MR 1068674 (91k:11063)
  • 10. D. Y. Kleinbock and G. A. Margulis.
    Flows on homogeneous spaces and Diophantine approximation on manifolds.
    Ann. of Math. (2), 148(1):339-360, 1998. MR 1652916 (99j:11083)
  • 11. Dmitry Kleinbock and Barak Weiss.
    Dirichlet's theorem on Diophantine approximation and homogeneous flows.
    Journal of Modern Dynamics (JMD), 2(1):43-62, 2008. MR 2366229 (2008k:11078)
  • 12. Shahar Mozes and Nimish A. Shah.
    On the space of ergodic invariant measures of unipotent flows.
    Ergodic Theory Dynam. Systems, 15(1):149-159, 1995. MR 1314973 (95k:58096)
  • 13. Marina Ratner.
    On Raghunathan's measure conjecture.
    Ann. of Math. (2), 134(3):545-607, 1991. MR 1135878 (93a:22009)
  • 14. Marina Ratner.
    Raghunathan's topological conjecture and distributions of unipotent flows.
    Duke Math. J., 63(1):235-280, 1991. MR 1106945 (93f:22012)
  • 15. Wolfgang M. Schmidt.
    Diophantine approximation. Lecture Notes in Mathematics, 785, Springer, Berlin, 1980, x+299 pp. MR 568710 (81j:10038)
  • 16. Nimish A. Shah.
    Uniformly distributed orbits of certain flows on homogeneous spaces.
    Math. Ann., 289(2):315-334, 1991. MR 1092178 (93d:22010)
  • 17. Nimish A. Shah.
    Limit distributions of polynomial trajectories on homogeneous spaces.
    Duke Math. J., 75(3):711-732, 1994. MR 1291701 (95j:22022)
  • 18. Nimish A. Shah.
    Limit distributions of expanding translates of certain orbits on homogeneous spaces.
    Proc. Indian Acad. Sci. (Math. Sci.), 106:105-125, 1996. MR 1403756 (98b:22024)
  • 19. Nimish A. Shah.
    Limiting distributions of curves under geodesic flow on hyperbolic manifolds.
    Duke Math. J., 148(2):251-279, 2009. MR 2524496
  • 20. Nimish A. Shah.
    Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation.
    Invent. Math., 177:509-532, 2009. MR 2534098

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 22E40, 11J83

Retrieve articles in all journals with MSC (2010): 22E40, 11J83


Additional Information

Nimish A. Shah
Affiliation: Tata Institute of Fundamental Research, Mumbai 400005, India
Address at time of publication: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: nimish@math.tifr.res.in; shah@math.osu.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-09-00657-2
Keywords: Flow on homogeneous spaces, geometry of numbers, Dirichlet's theorem, Minkowski's theorem, Diophantine approximation, unipotent flows, Ratner's theorem
Received by editor(s): December 15, 2008
Published electronically: December 29, 2009
Additional Notes: This research was supported in part by Swarnajayanti Fellowship
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.