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Multiple ergodic averages and combinatorics

Final Report Summary - MEAC (Multiple ergodic averages and combinatorics)

1a. Executive Summary-Summary of Project Context and Objectives.
Research: Ergodic theory has a long history of interaction with other mathematical fields. During the last thirty years a particularly fruitful one has been with Ramsey theory, the branch of combinatorics that seeks to find highly organized patterns in sufficiently large combinatorial structures. The deep connections between these two fields were first manifested with the ergodic theoretic proof of Szemer\'edi's theorem on arithmetic progressions, given by Furstenberg Since then, tools from ergodic theory have been applied to produce first-rate results in combinatorics, such as the density version of the Hales Jewett theorem and the polynomial extension of Szemer\'edi's theorem, results that are not currently attainable by any other methods. On the other direction, the field of ergodic theory has tremendously benefited as well, since the problems of combinatorial
and number-theoretical nature have given a boost to the in depth study of recurrence and convergence problems. The connecting link between combinatorics and ergodic theory is that regularity properties of integer subsets with positive density correspond to multiple recurrence properties of measure preserving systems. One normally establishes the multiple recurrence properties needed by analyzing the limiting behavior of some multiple ergodic averages. The research part of the project was devoted to an in depth analysis of the limiting behavior of such averages with an intention on deriving interesting combinatorial and number theoretic consequences.

Reintegration: The researcher has been fully integrated at the university of Crete. During he past 4 years the researcher has taught various undergraduate and graduate classes, has helped in the organization of the weekly analysis seminar, and has undertaken various other departmental responsibilities. He is currently an Assistant professor and is expected to be promoted to Associate professor at the beginning of 2014. The researcher believes that the working conditions at the university of Crete are excellent and make for a very fruitful research environment. The researcher has developed lasting co-operations with researchers from USA (the country from which I moved to Greece), has visited them and visited him several times during the last 4 years and have collaborated scientifically with some of them.

1b. Summary of Main Research Results
The performed research studied some central problems in ergodic theory that can be used to further advance the existing interaction between ergodic theory, combinatorics, and number theory. Overall, this was a multi-facet project and brought together tools from several diverse fields, such as ergodic theory, probability theory, combinatorics, analytic number theory, and Lie group theory. The results should be of interest not only to ergodic theorists but to people working in probability, combinatorics, and number theory.

In ergodic theory, the investigator, building on his previous work, carried out an in depth analysis of the limiting behavior of multiple ergodic averages along various integer sequences, for example Hardy sequences, random sequence, and sequences related to the prime numbers. The tools used include recent advances in the theory of characteristic factors and equidistribution results on nilmanifolds. The combinatorial implications were related to exhibiting patterns that can be found within every set of integers with positive density, thus obtaining several far reaching extensions of the celebrated theorem of Szemeredi on arithmetic progressions.

An additional pleasant surprise was that the performed research produced applications in directions of arithmetic that had not yet been explored, in particular in problems related to the partition regularity of non-linear equations in three variables (see article [8] below). This direction of research was largely unexplored and the tools developed leave much hope for further important advancements in combinatorics and number theory.

The work in line with the research objectives detailed in Part B of Annex I of the grant agreement,
has resulted in 9 publications/preprints that are detailed below. Some of these constitute partial
progress or a complete solution of Problems 1, 2, 3, 6 of Part B of Annex I of the grant agreement.
The list of publications follows and can be accessed in my research webpage
http://www.math.uoc.gr/~nikosf/publications.html

1. Ergodic averages of commuting transformations with distinct degree polynomial iterates.
Joint with C. Chu and B. Host. Proceedings of the London Mathematical Society,
102, (2011), 801-842.

2. Pointwise convergence for cubic and polynomial ergodic averages of non-commuting transformations
Joint with C.Chu. Ergodic Theory & Dynamical Systems, 32, no. 3, (2012), 877-897.

3. Random sequences and pointwise convergence of multiple ergodic averages.
Joint with E. Lesigne and M. Wierdl.Indiana University Mathematics Journal, 61, (2012), 585-617.

4. The polynomial multidimensional Szemeredi theorem along shifted primes.
Joint with B. Kra and B. Host. Israel Journal of Mathematics, 194, no. 1, (2013), 331-348.

5. Some open problems on multiple ergodic averages.

6. A multidimensional Szemeredi theorem for Hardy sequences of different growth.
To appear in Transactions of the American Mathematical Society.

7. Multiple recurrence for non-commuting transformations along rationally independent polynomials.
Joint with P. Zorin-Kranich. To appear in Ergodic Theory & Dynamical Systems.

8. Uniformity of multiplicative functions and partition regularity of some quadratic equations.
Joint with B. Host. Submitted for publication.

9. Random differences in Szemeredi's theorem and related results.
Joint with E. Lesigne and M. Wierdl. Submitted for publication.

Please see below for more details: Part 3 for the project objectives and Part 4
for the highlights of the work progress and achievements during this first period.