Some open problems on multiple ergodic averages (version of 2016)
Progress since 2016 (Latest update: April 2025.)
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 4. 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.
Journal d'Analyse Mathematique, 154, (2024), 199-253.
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 10.
Some progress made by A. Koutsogiannis.
Multiple ergodic averages for variable polynomials.
Discrete and Continuous Dynamical Systems,
42, (2022), 4637-4668.
Problem 11.
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 12.
2nd part: When the weight is the Mobius or the Liouville function solved by J. Teräväinen.
Pointwise convergence of ergodic averages with Mobius weight.
Preprint (2024).
Problem 13. Solved by A. Le for the first two sequences in
Nilsequences and multiple correlations along subsequences.
Ergodic Theory & Dynamical Systems,
40, (2020), no 6, 1634-1654.
and for the third sequence in
Interpolation sets and nilsequences.
Colloq. Math.,
162, (2020), 181-199.
Problem 15. Solved by N. F. and B. Kuca.
Joint ergodicity for commuting transformations and applications to polynomial sequences.
Inventiones Mathematicae,
239, (2025), 621-706.
Problem 16. Solved by N. F. and B. Kuca.
Joint ergodicity for commuting transformations and applications to polynomial sequences.
Inventiones Mathematicae,
239, (2025), 621-706.
Problem 17. Solved by N. F. and B. Kuca.
Joint ergodicity for commuting transformations and applications to polynomial sequences.
Inventiones Mathematicae,
239, (2025), 621-706.
Problem 19. 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 20. Solved by J.~Leng.
Structured extensions and multi-correlation sequences.
Preprint (2025). (The problem for polynomial multi-correlation sequences remains open.)
Problem 22. 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.
Some progress made by F.~Richter.
Uniform distribution in nilmanifolds along functions from a Hardy field..
Journal d'Analyse Mathématique,
149, (2023), 421-483.
More progress made by K.~Tsinas.
Pointwise convergence in nilmanifolds along smooth functions of polynomial growth..
Ergodic Theory & Dynamical Systems,
44, (2024), no 7, 1963-2008.
Problem 23.
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 25. 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 27. 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. Some progress made by L.~Daskalakis (covered c in (1,23/22)).
Ergodic theorems for bilinear averages, Roth's Theorem and Corners along fractional powers.
Preprint (2025).
Problem 30. 1st part. Some progress made by L.~Daskalakis (covered l=2 and c in (1,23/22)).
Ergodic theorems for bilinear averages, Roth's Theorem and Corners along fractional powers.
Preprint (2025).
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 34. Solved by N. F., O. Klurman, J. Moreira.
Partition regularity of Pythagorean pairs.
Preprint (2023).