An ergodic theoretic approach to Szemerédi's theorem [Thesis]

Description

Part I: Furstenberg Multiple Recurrence and Szemerédi's Theorem Chapter 1. Introduction 1. Ramsey Theory and Szemerédi's Theorem 2. A Non-Technical Overview of the Proof Chapter 2. Preliminaries and Notation 1. Preliminary Definitions 2. Notation and Conventions Chapter 3. The Furstenberg Multiple Recurrence Theorem and Szemerédi's Theorem 1. The Furstenberg Multiple Recurrence Theorem implies Szemerédi's Theorem 2. Szemerédi's Theorem implies the Furstenberg Multiple Recurrence Theorem Appendix 3.A. Ancillary Results for the Proof of Theorem 3.4 Appendix 3.B. Ancillary Results for the Proof of Theorem 3.6 Part II: Special Cases of the Furstenberg Multiple Recurrence Theorem Chapter 4. Weak Mixing Systems 1. Modes of Convergence 2. SZ Systems 3. Weak Mixing Systems are SZ Systems Chapter 5. Compact Systems 1. Almost Periodic Functions 2. Compact Systems are SZ Systems Appendix 5.A. Ancillary Results for the Proof of Theorem 5.6 Chapter 6. The Dichotomy Between Weak Mixing and Compact Systems 1. Factors and Extensions 2. Compact Systems and the Kronecker Factor 3. The Dichotomy of Systems Result Appendix 6.A. AP(X) is a Closed Subspace of L^2(X) Appendix 6.B. Ancillary Results for the Proof of Proposition 6.16 Part III: Extending the Special Cases Towards the Final Result Chapter 7. Further Preliminaries 1. Conditional Expectations 2. Hilbert Modules 3. Conditional Inner Products Chapter 8. Roth's Theorem Chapter 9. Weak Mixing Extensions 1. Relativising Weak Mixing Systems 2. The SZ property is Carried Through Weak Mixing Extensions Appendix 9.A. Ancillary Results for the Proof of Theorem 9.11 Chapter 10. Compact Extensions 1. Relativising Compact Systems 2. The SZ property is Carried Through Compact Extensions Appendix 10.A. Ancillary Results for the Proof of Theorem 10.12 Chapter 11. The Dichotomy Between Weak Mixing and Compact Extensions 1. The Relative Kronecker Factor 2. The Dichotomy of Extensions Result Appendix 11.A. AP(X|Y) is a Closed Subspace of L^2(X|Y) Appendix 11.B. Ancillary Results for the Proof of Theorem 11.8 Chapter 12. Furstenberg Towers and the Structure Theorem 1. The Existence of Furstenberg Towers 2. Limit Ordinals and the Final Conclusion Appendix 12.A. Generating -Algebras Appendix 12.B. Ancillary Results for the Proof of Theorem 12.7 Appendix A. Properties of Upper and Lower Density 1. Basic Properties of Upper and Lower Density 2. Properties of Syndetic Sets Appendix B. Properties of Cesàro and Density Convergence 1. Properties of Density and Cesàro Limits 2. Miscellaneous Convergence Results 3. Hierarchy of Density, Strong Cesàro and Cesàro Convergence Appendix C. Equivalent Formulations of Compact Systems Appendix D. Definitions of Function Spaces 1. L^p Spaces 2. Essentially Bounded Functions 3. Continuous Functions 4. The Weak Operator Topology Bibliography