Teoria ergódica: uma introdução e aplicações à teoria dos números.

• Ano de defesa: 2008
• Autor(a) principal: Lima, Yuri Gomes
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Palavras-chave em Português: Teorema de Birkhoff; Teorema de Van der Waerden; Teorema de Szemerédi; Teorema de Green-Tao; Birkhoff’s theorem; Van der Waerden’s theorem; Szemer´edi’s theorem; Green-Tao’s theorem
• Link de acesso: http://www.repositorio.ufc.br/handle/riufc/31838

In 2006, Ben Green and Terence Tao proved that the prime numbers contain arbitrarily large arithmetic progressions. The ideas of the proof come from Ergodic Theory and Harmonic Analysis and, since then, Ergodic Theory has attracted attention from the scientific community, because of its large interaction with other fields of Mathematics. This work develops this perspective, beginning from basic Ergodic Theory and arising at applications to Number Theory. Chapter 1 develops the necessary results from Measure Theory, introduces the basic examples and fixes notations. In Chapter 2, we prove the main tool to what follows, which is Birkhoff’s Ergodic Theorem. As a consequence of this theorem, we obtain many properties about the decimal expansion and continued fractions of the real numbers. Chapter 3 relates Ergodic Theory and Number Theory, and the main result here presented is Van der Waerden’s Theorem. This is a purely combinatorial result. Here, it will be proved as a consequence of the Multiple Birkhoff Theorem. In the end of this work, we sketch the main results in the area.