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
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 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 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
's containeod in
.
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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:
https://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.