Abstract:

It is a well known fact  that most of the Diophantine properties of a real number $\alpha$ are determined by its regular continued fraction expansion.In particular, from continued fraction we obtain all the best approximations to $\alpha$.In my talk, I will discuss  the behaviour of  best approximations in higher dimensional problems related to approximation of irrational linear subspaces in  $\mathbb{R}^d$  by integer vectors. It happened, that rational subspaces generated by the best approximations may have rather unusual properties.We will consider some  geometric observations including the phenomenon of degenerate dimension, behaviour of irrationality measure functions, definitions and properties of Diophantine exponents.

Video： http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Geometry_of_Multidimensional_Diophantine_approximation_and_Diophantine_exponents.html