Friendly measures, homogeneous flows and singular vectors

Singular Vectors on Manifolds over totally real Number Fields

We extend the notion of singular vectors in the context of Diophantine approximation of real numbers with elements of a totally real number field $K$. For $m\geq1$, we establish a version of Dani's correspondence in number fields and prove that under a class of `friendly measures' in $K_S^m$, the set of singular vectors has measure zero. Here $S$ is the set of Archimedean valuations of $K$ and $K_S$ is the product of the completions of $σ(K)$, $σ\in S$. On the other hand, we show the existence of uncountably many non-trivial singular vectors on suitable submanifolds of $K^m_S$ under the action…

Divergent trajectories on products of homogeneous spaces

In this paper, we determine the Hausdorff dimension of the set of points with divergent trajectories on the product of certain homogeneous spaces. The flow is allowed to be weighted with respect to the factors in the product space. The result is derived from its counterpart in Diophantine approximation. In doing this, we introduce a notion of jointly singular matrix tuples, and extend the dimension formula for singular matrices to such matrix tuples.

某些齐性空间乘积上锥作用的奇点

Singular points for cone actions on the product of certain homogeneous spaces

研究某些齐性空间乘积上锥作用的发散轨道，引入本质奇点概念，通过特定设置得出当\(\epsilon\in (0,\frac{1}{64})\)时本质发散轨道点集豪斯多夫维数的范围，扩展了前人在高维锥作用方面的结果。

Schmidt's Game on Certain Fractals

We construct (α,β) and α-winning sets in the sense of Schmidt's game, played on the support of certain measures (very friendly and awfully friendly measures) and show how to derive the Hausdorff dimension for some. In particular we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of R^N satisfying the open set condition then for any countable collection of non-singular affine transformations Λ_i:R^N \to R^N, dimK=dimK\cap (\cap ^{\infty}_{i=1}(Λ_i(BA))) where BA is the set of badly approximable vectors in R^N.