Geometry of Multidimensional Diophantine approximation and Diophantine exponents

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.

About the speaker

Professor Nikolay Moshchevitin received his Ph.D. in 1994 from Moscow State University in Russia. Since 2011 he is one of the managing editors of the Moscow Journal of Combinatorics and Number Theory. He works mostly in Diophantine Approximation, Geometry of Numbers and related topics. Among his students there are such known mathematicians as Andrew Raigorodskii, Oleg German and Ilya Shkredov. He solved several well-known problems, in particular he solved Wolfgang Schmidt's problems on the behavior of successive minima of a one-parameter family of lattices and constructed a counter-example to another Schmidt's problem related to approximation with positive integers. He will be teaching an introductory course in Diophantine Approximation.