The course gives a brief introduction into modern theory of Diophantine Approximation.Together with classical and basic facts we will deal with some recent results in the topic and discuss some unsolved problems. The course is devoted to several important one-and multi-dimensional problems in Diophantine Approximation We will study different continued fractions algorithms, methods of Geometry of Numbers, metrical Number Theory and related topics. Among the problems under consideration we deal with Diophantine spectra, uniform approximation, irrationality measure functions and approximation in higher dimensions.
Standard undergraduate courses in calculus, linear algebra and basic course in elementary Number Theory.
1. J. W. S. Cassels, An introduction to Diophantine approximation, 1957
2. W. M. Schmidt, Diophantine Approximation, 1980
3. T.W. Cusick, M.E. Flahive, The Markoff and Lagrange spectra, 1989
4. A.M. Rockett, P. Szusz, Continued fractions, 1992.
5. O.N. German, Geometry of Diophantine exponents, Preprint available at arXiv:2210.16553 (2022)
6. N. Moshchevitin, On some open problems in Diophantine Approximation, Preprint available at arXiv:1202.4539(2012)
7. N. Moshchevitin, Khintchine singular systems and their applications Russian Math. Surveys 65 (2010), no. 3, 433–511.
Target Audience: Undergraduate students, Graduate students
Teaching Language: English