题目: Badly approximable grids and k-divergent lattices 

报告人: Anurag Rao，北京大学

时间: 4月11日 16:00-17:00

地点: 5206

摘要: For a vector θ ∈ R^m we consider the resulting set of badly approximable targets Bad(θ). These are the points in the standard torus T^m which are badly approximated by the sequence of points qθ mod Z^m where q varies in the integers. When θ is nonsingular this set is known to have zero Lebesgue measure and full Hausdorff dimension. We study the intersection of Bad(θ) with cosets of closed subgroups in T^m and prove that, barring a certain topological condition on θ, Bad(θ) is necessarily null with respect to the Lebesgue measure on the coset. This topological condition on θ is called k-divergence and extends the classical notion of singularity. Moreover, to show our analysis is precise, we give an example of a k-divergent vector θ for which Bad(θ) contains the coset of a codimension 2 subgroup in Tm. Based on joint works with G. Lachman, N. G. Moshchevitin, U. Shapira and Y. Yifrach.