Some open problems on multiple ergodic averages (old version 2011)

  • Introduction.
  • Some useful tools and observations.
  • Some useful notions.
  • Problems related to general sequences.
  • Problems related to polynomial sequences.
  • Problems related to Hardy sequences.
  • Problems related to random sequences.
  • Extended bibliography organized by topic.

  • Global file (contains all problems).

  • Some slides (expanded version of a talk).

    Progress (Latest update: August 2024.)

    Problem numbering refers to this file.

  • Problem 1. A negative answer when the nilsequences N_x are defined by continuous functions by J. Briet and B. Green. The original version of the problem remains open.
        Multiple correlation sequences not approximable by nilsequences.
        Ergodic Theory & Dynamical Systems, 42, no. 9, (2022), 2711-2722.

  • Problem 2. The case where all sequence are equal and have linear growth solved by N. F.
        Multiple correlation sequences and nilsequences.
        Inventiones Mathematicae, 202, no. 2, (2015), 875-892.

  • Problem 3. Some progress made (solved for sequences that are good for seminorm control) by N. F. and B. Kuca.
        Degree lowering for ergodic averages along arithmetic progressions.
        To appear in Journal d'Analyse Mathematique.

  • Problem 5. Solved for l=2 by J. Griesmer.
        A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence.
        Ergodic Theory & Dynamical Systems, 44, no. 6, (2024), 1541-1580.

  • Problem 5. Solved for sequences of linear growth by N. F.
        Multiple correlation sequences and nilsequences.
        Inventiones Mathematicae, 202, no. 2, (2015), 875-892.

  • Problem 10. 1st part: For distal systems solved by W. Huang, S. Shao, X. Ye.
        Pointwise convergence of multiple ergodic averages and strictly ergodic models.
        Journal d'Analyse Mathematique, 139, no. 1, (2019), 265-305.

        2nd part: Solved by B. Krause, M. Mirek, and T. Tao.
        Pointwise ergodic theorems for non-conventional bilinear polynomial averages.
        Annals of Mathematics, 195, (2022), no 3, 997-1109.

  • Problem 11. Solved with an error term that is small (but not zero) in uniform density by N. F.
        Multiple correlation sequences and nilsequences.
        Inventiones Mathematicae, 202, no. 2, (2015), 875-892.

        For powers of the same transformation the non-ergodic case solved by A. Leibman.
        Nilsequences, null-sequences, and multiple correlation sequences.
        Ergodic Theory & Dynamical Systems, 35, (2015), no 1, 176–191.

  • Problem 12. Solved by M. Walsh.
        Norm convergence of nilpotent ergodic averages.
        Annals of Mathematics, 175, (2012), no 3, 1667-1688.

        An alternate ergodic proof is given by T. Austin in Int. Math. Res. Not. , 15, (2015), 6661-6674.

  • Problem 13. Solved by N. F. and B. Kuca.
        Joint ergodicity for commuting transformations and applications to polynomial sequences.
        Preprint (2022).

  • Problem 14. Solved by N. F. and B. Kuca.
        Joint ergodicity for commuting transformations and applications to polynomial sequences.
        Preprint (2022).

  • Problem 15. Solved by N. F. and B. Kuca.
        Joint ergodicity for commuting transformations and applications to polynomial sequences.
        Preprint (2022).

  • Problem 16. For distal actions solved by S. Donoso and W. Sun.
        Pointwise convergence of some multiple ergodic averages.
        Advances in Mathematics, 330, (2018), no 3, 946-996.

  • Problem 17. Solved by M. Walsh.
        Norm convergence of nilpotent ergodic averages.
        Annals of Mathematics, 175, (2012), no 3, 1667-1688.

        An alternate ergodic proof is given by T. Austin in Int. Math. Res. Not. 15 (2015), 6661-6674.

  • Problem 18. The case of rationally independent polynomials solved by N. F. and P. Zorin-Kranich.
        Multiple recurrence for non-commuting transformations along rationally independent polynomials.
        Ergodic Theory & Dynamical Systems 35, no. 2, (2015), 403-411.

  • Problem 19. Corrected statement: The collection of sequences is good for l-convergence of a single transformation if and only if it is good for l-convergence of rotations on the torus.

  • Problem 20. Some progress made by V. Bergelson, J. Moreira, F. Richter.
        Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications.
        Advances in Mathematics, 443, 109597 (2024), (50pp) (Preprint appeared in 2020)

        Some progress made by N. F.
        Joint ergodicity of sequences.
        Advances in Mathematics, 417, 108918 (2023), (63pp).

        Solved by K. Tsinas.
        Joint ergodicity of Hardy field sequences.
        Transactions of the American Mathematical Society. 376, (2023), 3191-3263.

  • Problem 21. Solved by K. Tsinas.
        Joint ergodicity of Hardy field sequences.
        Transactions of the American Mathematical Society,376, (2023), 3191-3263.

  • Problem 22. Solved by V. Bergelson, J. Moreira, F. Richter.
        Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications.
        Advances in Mathematics, 443, 109597 (2024), (50pp).

  • Problem 23. Solved by V. Bergelson, J. Moreira, F. Richter.
        Single and multiple recurrence along non-polynomial sequences.
        Advances in Mathematics, 368, (2020), 107-146.

  • Problem 25. Some progress made by N. F. (reduction to nilsystems)
        Joint ergodicity of fractional powers of primes.
        Forum of Mathematics, Sigma, 10, (2022), e30.

        Solved by A. Koutsogiannis and K. Tsinas.
        Ergodic averages for sparse sequences along primes.
        Preprint (2023).

  • Problem 29. Solved by N. F.
        A multidimensional Szemeredi theorem for Hardy sequences of different growth.
        Transactions of the American Mathematical Society, 367, no. 8, (2015), 5653-5692.

  • Problem 31. Some progress made by N. F., E. Lesigne, M. Wierdl.
        Random differences in Szemeredi's theorem and related results.
        Journal d'Analyse Mathematique. 130, no. 1, (2016), 91-133.

        More progress made by J. Briet, Z. Dvir, S. Gopi.
        Outlaw distributions and locally decodable codes.
        Theory of Computing. 15, (2019), 1-24.

        More progress made by J. Briet, S. Gopi.
        Gaussian width bounds with applications to arithmetic progressions in random settings.
        International Mathematics Research Notices. 2020, no. 22, (2020), 8673-8696.

        More progress made by J. Briet, D. Castro-Silva.
        On the threshold for Szemeredi's theorem with random differences.
        Preprint (2023)

  • Problem 32. Some progress made by N. F., E. Lesigne, M. Wierdl.
        Random differences in Szemeredi's theorem and related results.
        Journal d'Analyse Mathematique. 130, no. 1, (2016), 91-133.