Dirichlet’s theorem on Diophantine approximation

4:00 pm Thursday, October 15, 2009
Nimish A. Shah (Ohio State University)

In the late sixties, Baker and Schmidt proved that Dirichlet’s theorem on simultaneous Diophantine approximation of n real numbers x_1,...,x_n$ cannot be improved in a certain precise sense for almost all (x_1,...,x_n) in R^n. Answering a question asked by them, we show that the non-improvability result also holds for almost all points on any analytic curve in R^n which is not contained in a proper affine subspace. The problem is reformulated in terms of a dynamical question on homogeneous spaces which is then resolved by proving a new equidistribution theorem for on the space of unimodular lattices in R^{n+1}. The proof involves Ratner's theorem on unipotent flows and linearization technique.

 

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