报告题目：Diophantine approximations

报 告 人：Nikolay Moshchevitin 教授 Moscow State University

报告地点：ZOOM ID：894 9789 1576； Password：237198

Lecture 1. 2022.01.19， 13：30-15：00，Continued fractions.

Representation of real numbers as continued fractions. Approximation by convergents, Perron's formula.
Irrationality measure function.
Lagrange and Dirichlet spectra. Minkowski diagonal continued fraction and the related spectrum. Geometry of continued fractions algorithm.

Lecture 2. 2022.01.20， 13：30-15：00，Distribution of Farey fractions.

Elements of basic number theory. Euler and Moebius functions.
Integral representation of greatest common divisor. Counting the number of reduced fractions in [0,1].
Fraenel's theorem. Relation to Riemann zeta-function.

Lecture 3. 2022.01.26， 13：30-15：00，Farey tree and Minkowski function.

Stern-Brocot sequences. Minkowski question-mark function.
Salem's theorem on the derivative. Fixed points problem. An analog to Fraenel's theorem.
Fourier-Stieltjes coefficients.

Lecture 4. 2022.01.27， 13：30-15：00，Multidimensional approximation.

Simultaneous approximation to real numbers. Linear forms close to xero.
Basic laws of approximation. Best approximations and phenomenon of degeneracy of dimension.

Diophantine exponents and Jarnik's inequalities.

报告人简介：Nikolay Moshchevitin教授现就职于莫斯科国立大学，研究兴趣为Geometry of numbers, geometric theory of Diophantine approximations, ergodic and combinatorial theory of numbers, theory of dynamical systems。共发表论文70余篇，超过400次引用。于1998年获得俄罗斯联邦国家奖。