Ergodic Ramsey Theory

Master Lecture: Department of Mathematics, SNSB (Scoala Normala Superioara Bucuresti), Winter Semester 2010/2011

Lecturer: Laurentiu Leustean

Time and location: Tuesday, 10:00-14:00, Lecture Hall Grigore Moisil (412), IMAR

Lecture Notes (version 31.01.2011)

Ergodic Ramsey theory was initiated in 1977 when Hillel Furstenberg proved a far reaching extension of the classical Poincare recurrence theorem and derived from it the celebrated Szemeredi's theorem, which states that any subset of integers of positive upper density must necessarily contain arbitrarily long arithmetic progressions.

Since then, Furstenberg's ergodic approach was used to establish many more types of recurrence theorems, which (via the Furstenberg's correspondence principle) yield a number of highly non-trivial combinatorial theorems. Many of the results obtained by these ergodic techniques are not known, even today, to have any "elementary" proof, thus testifying to the power of this method.

Lectures :

Lecture 1: a general presentation of the course.

Lecture 2: Topological Dynamical Systems: definitions, examples. Basic constructions: homomorphisms, (strongly) invariant sets.

Lecture 3: Basic constructions continued: subsystems, direct products, disjoint unions. Transitivity.

Lecture 4: Minimality. Recurrence. Application to a result of Hilbert, presumably the first result of Ramsey Theory.

Lecture 5: Multiple Recurrence Theorem.

Lecture 6: Ramsey Theory: van der Waerden Theorem.

Lecture 7: Ultrafilter approach to Ramsey Theory.

Lecture 8: Hales-Jewett Theorem. Measure-preserving systems.

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

Lecture 10: Different notions of density. Furstenberg correspondence principle.

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

Lecture 12: Mean ergodic theorem in uniformly convex Banach spaces. Finitary version.

Lecture 13: Recurrence. Szemeredi theorem - finitary and ergodic versions.

Lecture 14: Mixing. Szemeredi property for compact and weak mixing systems.

Lecture 15: Proof of Roth Theorem.

Seminars and Homeworks:

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

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

Homework Sheets: [1], [2], [3], [4], [5]

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
• Terence Tao, Ultrafilters, nonstandard analysis, and epsilon management
• William T. Gowers, The two cultures of mathematics
• Terence Tao, Roth's Theorem.
• Akos Magyar, Topics in ergodic theory.

Papers:

• Hillel Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi 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
• Philipp Gerhardy, Proof mining in topological dynamics, Notre Dame Journal of Formal Logic, vol. 49, no. 4, pp. 431-446 (2008)
• 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 Szemerédi 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.
• Ulrich Kohlenbach, Laurentiu Leustean, A quantitative Mean Ergodic Theorem for uniformly convex Banach spaces , Ergodic Theory and Dynamical Systems 29 (2009), 1907-1915; Erratum : Vol. 29 (2009), No. 6, 1995.
• Jeremy Avigad, Philipp Gerhardy, Henry Towsner, Local stability of ergodic averages, Transactions of the AMS 362 (2010), 261-288.
• Terence Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory and Dynamical Systems 28 (2008), 657–688.
• Endre Szeméredi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hungar. 20 (1969), 89-104.
• Endre Szeméredi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199-245.
• Hillel Furstenberg, Yitzhak Katznelson, Donald S. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 527--552.
• Terence Tao, A quantitative ergodic theory proof of Szemerédi's theorem, The Electronic Journal of Combinatorics, R99, 2006.
• Klaus Roth, On certain sets of integers, J. London Math Soc. 28 (1953), 104-109.