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Hausdorff dimension estimates for bounded orbits on homogeneous spaces of Lie groups

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Interactions between Dynamics of Group Actions and Number Theory

This work is motivated by studying badly approximable vectors, that is, ${f x}in {f R}n$ such that $|q{f x} – {f p}| ge cq{-1/n}$ for all ${f p}in {f Z}^n$, $qin {f N}$. Computing the Hausdorff dimension of the set of such ${f x}$ for fixed $c$ is an open problem. I will present some estimates, based on the interpretation of a badly approximable vector via a trajectory on the space of lattices, and then use exponential mixing to estimate from above the dimension of points whose trajectories stay in a fixed compact set. Joint work with Ryan Broderick.

This talk is part of the Isaac Newton Institute Seminar Series series.

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