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

Shah, Nimish

doi:10.1090/S0894-0347-09-00657-2

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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
Posted: December 29, 2009
MathSciNet review: 2601043
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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 ${SL}(m,\mathbb{R})$ 's containeod in ${SL}(k+1,\mathbb{R})$ .

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