Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Journal of the American Mathematical Society
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

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?)


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
PII: S 0894-0347(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.