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

✍ Scribed by Walt van Amstel

• Year: 2019
• Tongue: English
• Leaves: 139
• Category: Library

✦ Table of Contents

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