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Master's Dissertation
DOI
Document
Author
Full name
Paulo Henrique de Souza Macedo Arruda
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2019
Supervisor
Committee
Franco Filho, Antonio de Padua (President)
Aurichi, Leandro Fiorini
Boero, Ana Carolina
Title in English
Topological equivalences of the Erds-Turán Conjecture
Keywords in English
Equivalences
Erds-Turán conjecture
Abstract in English
The main goal of this project was study a topological path for number theory, specially the Erds-Turán Conjecture on the sum of reciprocals which, at the present moment when this work is being concluded, lies without an answer. The machinery used was the semigroup structure and the dynamics of translations of the ech-Stone compactification of the natural numbers. For this end, here follows themes dear to Ramsey theory, combinatorics, ergodic theory and combinatorial number theory, likewise all topological and algebraic structures underlying the construction and manipulation of all objects hereon. The major result described in this document, due to N. Hindman, was equivalences of the aforementioned conjecture written within a topological framework. Other impor- tant results also discussed thereon with this framework was the Theorems of van der Waerden and Szemerédi.
Title in Portuguese
Equivalências topológicas da conjectura de Erds-Turán
Keywords in Portuguese
Conjectura de Erds-Turán
Equivalências
Abstract in Portuguese
O objetivo desse trabalho foi estudar caminhos topológicos para a teoria dos números, em especial a conjectura de Erds-Turán acerca da soma dos recíprocos que, no momento no qual este trabalho se conclui, encontra-se aberta. As ferramentas usadas foram a estrutura de semigrupo e a dinâmica que essa estrutura imprime à compactificação de ech-Stone dos números naturais por translação. Para tal objetivo, foram abordados temas caros à teoria de Ramsey, combinatória, teoria ergódica e teoria combinatória dos números, além da construção dos objetos topológicos e bem como suas estruturas algébricas. O principal resultado, atribuído a N. Hindman, é uma série de equivalências topológ- icas da conjectura supracitada. Outros resultados conhecidos, tais como os Teoremas de van der Waerden e Szemerédi são abordados com o mesmo olhar topológico.
 
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Publishing Date
2019-11-04
 
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