**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.