Invariant measures and their applications to number theory

• Grant number: 11/06796-6
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2011
• End date: June 30, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Benito Frazao Pires
• Grantee: Pedro Oliveira Mariz de Carvalho
• Host Institution: Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil
• Associated research grant: 09/02380-0 - Flows on surfaces and exchange transformations, AP.JP

Abstract

This research proposal is an introduction to ergodic theory focusing on its applications to number theory. More specifically, it consists in studying the stochastic behaviour of the following sequences of integers {1,2,4,8,1,3,6,1,2,5,1,...}(obtained taking the first digits of {2^n} and the sequence of the digits of the decimal expansion of a real number. In this way, the project lies divided into two parts: firstly, the studentis supposed to study measure and ergodic theory; and at a second moment the student will approachthe existence of invariant measures and its applications to number theory.