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

Shah, Nimish

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

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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