An ergodic theoretic approach to Szemerédi's theorem

Abstract

In this dissertation, Szemer edi's Theorem is proven using ergodic theoretic techniques via the Furstenberg Multiple Recurrence Theorem. Brief historical remarks, along with a non-technical layout of the ideas behind the proof of the Furstenberg Multiple Recurrence Theorem, are given in Chapter 1. After introducing some notation, preliminary de nitions and propositions in Chapter 2, the equivalence of the Furstenberg Multiple Recurrence Theorem and Szemer edi's Theorem is laid out in detail in Chapter 3. The rest of this work is devoted to providing a proof of the Furstenberg Multiple Recurrence Theorem. Two important classes of invertible measure preserving systems, weak mixing and compact systems, are introduced in Chapters 4 and 5 respectively, where it is shown that these classes of measure preserving systems satisfy the Furstenberg Multiple Recurrence Theorem. (We shall say these systems have the Furstenberg property). In Chapter 6, a dichotomy result is proven that characterizes all invertible measure preserving systems in terms of weak mixing and compact systems. After introducing more preliminary de nitions and propositions in Chapter 7, a short proof of Roth's Theorem, the rst non-trivial special case of Szemer edi's Theorem, is given in Chapter 8. In Chapter 9, a generalization of weak mixing systems, known as weak mixing extensions, is introduced. It is shown that if a measure preserving Y has the Furstenberg property and X is a weak mixing extension of Y, the Furstenberg property passes through the extension to the extended system X. The analogous generalization of compact systems - compact extensions - is introduced in Chapter 10 and it is shown that the Furstenberg property passes through compact extensions. Similar to what was done in Chapter 6, a dichotomy result is proven in Chapter 11 that characterizes extensions of invertible measure preserving systems in terms of weak mixing and compact extensions. All of the necessary tools developed in previous chapters are put to use in Chapter 12 where the Furstenberg Multiple Recurrence Theorem is proven - thus establishing Szemer edi's Theorem.