## Atividades

### Building bridges between Diophantine approximation, continued fractions and measure theory

Modalidade: Presencial.

Professor: Christian Sanabria Castaneda.

Duração: 08, 09 e 10 de janeiro de 2025.

Dias e horários: Quarta, quinta e sexta, das 14:00 às 16:00.

Público-alvo: Graduandos e pós-graduandos em matemática ou áreas afins.

Programa:

In this mini course we will show the interplay between three fascinating and apparently unrelated topics in Mathematics: Diophantine approximation, continued fractions and measure theory. Our exposition starts with the introduction of the concept of continued fraction and some of its basic and fundamental properties. This will lead us to a method for approximating real numbers using rational ones, creating a link with the theory of Diophantine approximation, proving some of its classical results. To extend the applicability of continued fractions,
we will adopt a functional approach to the sequence of partial quotients. By combining this with elements of Lebesgue measure, we will derive significant statistical insights of the real line from the continued fraction probability distribution. Some key results in this direction are the rate of growth of the sequence of partial quotients and their product, as well as bounds for the sequence of partial convergents for almost every real number. We will conclude by illustrating
how the interaction between measure theory and Diophantine approximation leads us to two major results in analytic number theory: the Khintchine approximation theorem, and the Duffin-Schaeffer (or Koukoulopoulos-Maynard) theorem.

Bibliografia:

1) J. Borwein, A. van der Poorten, J. Shallit, and W. Zudilin, Neverending fractions: an introduction to continued fractions. Cambridge University Press, 2014.

2) G. H. Hardy, E. M. Wright, D. R. Heath-Brown and J. H. Silverman, An introduction to the theory of numbers, sixth edition. Oxford University Press, 2009.

3) A. Y. Khinchin, Continued fractions, 1964.

4) I. Niven, On asymmetric Diophantine approximations. Michigan Mathematical Journal, 9(2), 121-123, 1962.

5) C. D. Olds, Continued fractions. The Mathematical Association of America, 1963.

### Contato

Instituto de Ciências Matemáticas e de Computação – ICMC

Av. Trabalhador São-carlense, 400 – Centro

São Carlos-SP, CEP: 13566-590

Email: verao@icmc.usp.br