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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms
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by Nimish A. Shah PDF
J. Amer. Math. Soc. 23 (2010), 563-589 Request permission

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 $\mathrm {SL}(m,\mathbb {R})$’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
  • Email: nimish@math.tifr.res.in; shah@math.osu.edu
  • Received by editor(s): December 15, 2008
  • Published electronically: December 29, 2009
  • Additional Notes: This research was supported in part by Swarnajayanti Fellowship
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2601043