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
DOI:
https://doi.org/10.1090/S0894-0347-09-00657-2
Published electronically:
December 29, 2009
MathSciNet review:
2601043
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $\mathrm {SL}(m,\mathbb {R})$’s containeod in $\mathrm {SL}(k+1,\mathbb {R})$.
- R. C. Baker, Dirichlet’s theorem on Diophantine approximation, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 37–59. MR 485713, DOI https://doi.org/10.1017/S030500410005427X
- Yann Bugeaud, Approximation by algebraic integers and Hausdorff dimension, J. London Math. Soc. (2) 65 (2002), no. 3, 547–559. MR 1895732, DOI https://doi.org/10.1112/S0024610702003137
- J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin-New York, 1971. Second printing, corrected; Die Grundlehren der mathematischen Wissenschaften, Band 99. MR 0306130
- S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55–89. MR 794799, DOI https://doi.org/10.1515/crll.1985.359.55
- S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 101 (1991), no. 1, 1–17. MR 1101994, DOI https://doi.org/10.1007/BF02872005
- S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 91–137. MR 1237827
- H. Davenport and W. M. Schmidt, Dirichlet’s theorem on diophantine approximation. II, Acta Arith. 16 (1969/70), 413–424. MR 279040, DOI https://doi.org/10.4064/aa-16-4-413-424
- H. Davenport and Wolfgang M. Schmidt, Dirichlet’s theorem on diophantine approximation, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 113–132. MR 0272722
- M. M. Dodson, B. P. Rynne, and J. A. G. Vickers, Dirichlet’s theorem and Diophantine approximation on manifolds, J. Number Theory 36 (1990), no. 1, 85–88. MR 1068674, DOI https://doi.org/10.1016/0022-314X%2890%2990006-D
- D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2) 148 (1998), no. 1, 339–360. MR 1652916, DOI https://doi.org/10.2307/120997
- Dmitry Kleinbock and Barak Weiss, Dirichlet’s theorem on Diophantine approximation and homogeneous flows, J. Mod. Dyn. 2 (2008), no. 1, 43–62. MR 2366229, DOI https://doi.org/10.3934/jmd.2008.2.43
- Shahar Mozes and Nimish Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 149–159. MR 1314973, DOI https://doi.org/10.1017/S0143385700008282
- Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, DOI https://doi.org/10.2307/2944357
- Marina Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235–280. MR 1106945, DOI https://doi.org/10.1215/S0012-7094-91-06311-8
- Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710
- Nimish A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), no. 2, 315–334. MR 1092178, DOI https://doi.org/10.1007/BF01446574
- Nimish A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J. 75 (1994), no. 3, 711–732. MR 1291701, DOI https://doi.org/10.1215/S0012-7094-94-07521-2
- Nimish A. Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 2, 105–125. MR 1403756, DOI https://doi.org/10.1007/BF02837164
- Nimish A. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Math. J. 148 (2009), no. 2, 251–279. MR 2524496, DOI https://doi.org/10.1215/00127094-2009-026
- Nimish A. Shah, Equidistribution of expanding translates of curves and Dirichlet’s theorem on Diophantine approximation, Invent. Math. 177 (2009), no. 3, 509–532. MR 2534098, DOI https://doi.org/10.1007/s00222-009-0186-6
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
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.
