## Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms

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- by Nimish A. Shah
- J. Amer. Math. Soc.
**23**(2010), 563-589 - DOI: https://doi.org/10.1090/S0894-0347-09-00657-2
- Published electronically: December 29, 2009
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## 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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## Bibliographic 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