Topological equivalences of the Erds-Turán Conjecture
v Agradecimentos vi Acknowledgements vii Bibliography 81 Index 85 ii The life of Aristotle may be resumed in one single sentence: Aristotle was born, he thought and he died. Everything else that could be said on the matter is pure anedocte. Martin Heidegger; as cited by J. Derrida in K. Dick and A. Z. Kofman (2002) Derrida. Asking whether and how a proposition can be verified is only a particular way of asking "How d'you mean?". The answer is a contribution to the grammar of the proposition.
... the proposition. Ludwig Wittgenstein; in [Wit58] Human beings do not live in the objective world alone, nor alone in the world of social activity as ordinarily understood, but are very much at the mercy of the particular language which has become the medium of expression for their society. It is quite an illusion to imagine that one adjusts to reality essentially without the use of language and that language is merely an incidental means of solving specific problems of communication or reflection. The fact of the matter is that the "real world" is to a large extent unconsciously built upon the language habits of the group. Edward Sapir; in [Sap29] Zeno was concerned, as a matter of fact, with three problems, each presented by motion, but each more abstract than motion, and capable of a purely arithmetical treatment. These are the problems of the infinitesimal, the infinite, and continuity. To state clearly the difficulties involved, was to accomplish perhaps the hardest part of the philosopher's task. This was done by Zeno. From him to our own day, the finest intellects of each generation in turn attacked the problems, but achieved, broadly speaking, nothing. In our own time, however, three men -Weierstrass, Dedekind, and Cantor -have not merely advanced the three problems, but have completely solved them. Russell, Bertrand; in Mathematics and the metaphysician, [Rus08]. iii Resumo O objetivo desse trabalho foi estudar caminhos topológicos para a teoria dos números, em especial a conjectura de Erdős-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 Čech-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 topológicas da conjectura supracitada. Outros resultados conhecidos, tais como os Teoremas de van der Waerden e Szemerédi são abordados com o mesmo olhar topológico. iv Abstract The main goal of this project was study a topological path for number theory, specially the Erdős-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 Čech-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 important results also discussed thereon with this framework was the Theorems of van der Waerden and Szemerédi. v Agradecimentos Em primeiro lugar, gostaria de agradecer a todos os meus professores, desde o primário até a pós-graduação; sem eles, o presente trabalho não seria possível. Entre esses, um agradecimento especial ao Prof. Dr. Antônio de Pádua Franco Filho, meu orientador, pelo suporte e confiança durante o mestrado, e a Ana Paula Sanches por ter possibilitado a conclusão desse trabalho. Esse trabalho é dedicado a todos meus amigos, em especial