Analysis and Number Theory
Instructor: Nikolay Moshchevitin
Course Introduction:
We will study how real numbers can be approximated by rational numbers, more generally, how to approximate complex objects (such as arbitrary linear subspaces of Euclidean space) by simple object (rational subspaces). This topic is known as Diophantine Approximation.
We will start with an introductory lecture where we explain the very basic concept of Diophantine Approximation and easiest application, and then discuss in more details various one-dimensional and multi-dimensional problems.
A lot of time we will devote to the theory of continued fractions, that is representation of numbers in a form $$ [b_0;b_1,\dots,b_s,\dots]= b_0+ \frac{1}{\displaystyle{b_1+\frac{1}{\displaystyle{b_2 + \frac{1}{\displaystyle{b_3 +\dots + \displaystyle{\frac{1}{b_{s}+\dots}} }}}}}} . $$
Representation of reals by continued fraction allows to study various arithmetical properties of real numbers. We deal with general approximation laws, irrationality measure functions and study some classical and modern results. Beginning with simpler objects as Farey fractions and Stern-Brocot sequences, related to the procedure of constructing all rational numbers in $[0,1]$ by means of the operation $$\frac{a}{b},\frac{c}{d}\,\,\, \mapsto \,\,\, \frac{a+c}{b+d}, $$ we will continue with Minkowski question mark function $?(x)$, nonlinear approximation and certain related problems. Some basic Number Theory (properties of Euler function $\varphi (n)$ an Möbius function $\mu(n)$, quadratic residues and non-residues, etc...) and Classical Analysis will be also involved. A part of the course is devoted to the problems of uniform distribution of sequences in $\mathbb{R}^d$.
In the second part of the course we will deal with objects related to Geometry of Numbers.
We will give geometrical interpretation for many objects and theorems from continuer fractions' theory and in particular explain Klein's geometrical construction.
Starting from Minkowski convex body theorem, we will discuss how geometric objects (lattices) may be useful for arithmetical problems (for example representation of integers as sums of two and four squares) and deal with some applications. At the end we will consider problems related to Minkowski's successive minima, uniform approximation and Diophantine exponents.