Diophantine exponents, best approximation and badly approximable numbers
Appears in collection : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire
We will discuss recent progress in analysis of uniform and ordinary Diophantine exponents $\hat\omega $ and $\omega$ for linear Diophantine approximation as well as some applications of the related methods. In particular, we give a new criterion for badly approximable vectors in $\mathbb{R}^{d}$ the behavior of the best approximation vectors in the sense of simultaneous approximation and in the sense of linear form. It turned out that compared to the one-dimensional case our criterion is rather unusual. We apply this criterion to the analysis of Dirichlet spectrum for simultaneous Diophantine approximation.
