Expanding Translates of Curves and Dirichlet-minkowski Theorem on Linear Forms
نویسنده
چکیده
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 on R which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in 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,R)’s contained in SL(k + 1,R).
منابع مشابه
Equidistribution of Expanding Translates of Curves and Dirichlet’s Theorem on Diophantine Approximation
We show that for almost all points on any analytic curve on R which is not contained in a proper affine subspace, the Dirichlet’s theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain parti...
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