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. |