Dimension estimates for the set of points with non-dense orbit in\n homogeneous spaces

2017/10/13 by Dmitry Kleinbock, Kleinbock, Dmitry, Shahriar Mirzadeh +1
Mathematics · #11J13 (Secondary) #37A17 #37A25 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1710.04898

openalex publication_date 2017/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X = G/\Γ, where G is a Lie group and \Γ is a lattice in\nG, and let U be a subset of X whose complement is compact. We use the\nexponential mixing results for diagonalizable flows on X to give upper\nestimates for the Hausdorff dimension of the set of points whose trajectories\nmiss U. This extends a recent result of Kadyrov and produces new applications\nto Diophantine approximation, such as an upper bound for the Hausdorff\ndimension of the set of weighted uniformly badly approximable systems of linear\nforms, generalizing an estimate due to Broderick and Kleinbock.\n

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