Ressource pédagogique : Alexander Gorodnik - Diophantine approximation and flows on homogeneous spaces (Part 1)

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The fundamental problem in the theory of Diophantine approximation is to understand how well points in the Euclidean space can be approximated by rational vectors with given bounds on denominators. It turns out that Diophantine properties of points can be encoded using flows on homogeneous spaces, and in this course we explain how to use techniques from the theory of dynamical systems to address some of questions in Diophantine approximation. In particular, we give a dynamical proof of Khinchin’s theorem and discuss Sprindzuk’s question regarding Diophantine approximation with dependent quantities, which was solved using non-divergence properties of unipotent flows. In conclusion we explore the problem of Diophantine approximation on more general algebraic varieties.

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