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

Shah, Nimish^1, 2

¹ Tata Institute of Fundamental Research, Mumbai 400005, India
² Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Journal of the American Mathematical Society, Tome 23 (2010) no. 2, pp. 563-589

Résumé

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 $\mathrm {SL}(k+1,\mathbb {R})$.

Détail

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

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