Ergodic Theory and Additive Combinatorics

Master Lecture: Department of Mathematics, SNSB (Scoala Normala Superioara Bucuresti), Summer Semester 2013

Lecturer: Laurentiu Leustean

Lecture Notes (version 05.07.2013)

Lectures :

Lecture 1: a general presentation of the course.

Lecture 2: Topological Dynamical Systems: definitions, examples.

Lecture 3: Basic constructions continued: homomorphisms, (strongly) invariant sets, subsystems, direct products, disjoint unions. Transitivity.

Lecture 4: Minimality. Recurrence.

Lecture 5: Application to a result of Hilbert, presumably the first result of Ramsey Theory.

Lecture 6: Multiple Recurrence Theorem.

Lecture 7: Ramsey Theory: van der Waerden Theorem.

Lecture 8: Ultrafilter approach to Ramsey Theory.

Lecture 9: Ergodic Theory: measure-preserving systems, induced operator.

Lecture 10: Bernoulli shift. Recurrence.

Lecture 11: Ergodicity. Maximal ergodic theorems. Birkhoff ergodic theorem

Seminars:

Seminar Sheets: [1], [2], [3], [4], [5], [6], [7], [8], [9]

Solutions: [1], [2], [3], [4], [5], [6], [7], [8], [9]

Useful links:

• Polymath Blog
• Ergodic Theory and Additive Combinatorics, Program at Mathematical Sciences Research Institute (MSRI), Berkeley, August 18, 2008 - December 19, 2008
• Summer School: Analysis and Ergodic Theory, September 17 -September 22, 2006, Lake Arrowhead, California

Books:

• Hillel Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981
• Ronald L. Graham, Bruce L. Rotschild, Joel H. Spencer, Ramsey Theory, John-Wiley & Sons, 1980.
• Randall McCutcheon, Elemental Methods in Ergodic Ramsey Theory, Springer, 1999
• Douglas A. Lind, Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, 1995
• Neil Hindmann, Donna Strauss, Algebra in the Stone-Čech compactification: theory and applications, Walter de Gruyter, 1998
• Peter Walters, An Introduction to Ergodic Theory, Springer, 2000
• Paul Halmos, Lectures on Ergodic Theory, Chelsea, 1956
• Ulrich Krengel, Ergodic Theorems, van Nostrand, 1975

Lecture notes, surveys, essays:

• Terence Tao, Ergodic Theory, in: Poincare's Legacies, Part I: pages from year two of a mathematical blog, AMS, 2009; a draft version can be downloaded here
• Ben Green, Ergodic Theory, lecture notes for a 2008 course at Cambridge University
• surveys by Vitaly Bergelson, Bryna Kra.
• Terence Tao, Soft analysis, hard analysis, and the finite convergence principle
• Terence Tao, The correspondence principle and finitary ergodic theory

Papers:

• Hillel Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, Journal d'Analyse Mathematique 31 (1977), 204-256
• Hillel Furstenberg, Benjamin Weiss, Topological dynamics and combinatorial number theory, Journal d'Analyse Mathematique 34 (1978), 61--85
• Vitaly Bergelson, Alexander Leibman, Polynomial extensions of van der Waerden's and Szemeredi's theorems , J. Amer. Math. Soc. 9 (1996), 725-753
• Saharon Shelah, Primitive recursive bounds for van der Waerden numbers, J. Amer. Math. Soc. 1 (1988), 683–697
• William T. Gowers, A new proof of Szemeredi's theorem, GAFA 11 (2001), 465-588.
• Neil Hindman, Finite sums from sequences within cells of a partition of N, J. Combinatorial Theory (Series A) 17 (1974), 1-11
• Alfred Hales, Robert Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229.