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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
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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 $\mathrm {SL}(k+1,\mathbb {R})$.

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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

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.