PSU Mark
Eberly College of Science Mathematics Department

Meeting Details

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Title:Hausdorff dimension of the set of Singular Vectors
Seminar:Center for Dynamics and Geometry Seminars
Speaker:Yitwah Cheung (SFSU)
Abstract:
Singular vectors in R^d correspond to divergent trajectories of the homogeneous flow on SL(d+1,R)/SL(d+1,Z) induced by the one parameter subgroup diag(e^t,...,e^t,e^{-dt}) acting by left multiplication. In this talk, I will sketch a proof of the following result: the Hausdorff dimension of the set of singular vectors in R^2 is 4/3. (Alternatively, the set of points lying on divergent trajectories of the homogeneous flow on SL(3,R)/SL(3,Z) has Hausdorff dimension 7 and 1/3.) The main idea involves a multi-dimensional generalisation of continued fraction theory from the perspective of the best approximation properties of convergents. As an application, we answer a question of A.N. Starkov regarding the existence of slowly divergent trajectories.

Room Reservation Information

Room Number:MB106
Date:11 / 05 / 2007
Time:03:35pm - 05:40pm