# Progress since 2016 (Latest update: January 2023.)

## 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.
To appear in Ergodic Theory & Dynamical Systems.

• 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.
Preprint.

• 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.
Preprint.

• 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 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.
Preprint.

• Problem 16. Solved by N. F. and B. Kuca.
Joint ergodicity for commuting transformations and applications to polynomial sequences.
Preprint.

• Problem 17. Solved by N. F. and B. Kuca.
Joint ergodicity for commuting transformations and applications to polynomial sequences.
Preprint.

• 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 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.

• 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.
Preprint.

Some progress made by N. F.
Joint ergodicity of sequences.
To appear in Advances in Mathematics.

Solved by K. Tsinas.
Joint ergodicity of Hardy field sequences.
To appear in the Transactions of the American Mathematical Society.

• Problem 25. Solved by V. Bergelson, J. Moreira, F. Richter.
Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications.
Preprint.

• 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.

• Problem 31. Some progress made by J. Briet, Z. Dvir, S. Gopi.
Outlaw distributions and locally decodable codes.
Theory of Computing. 15, (2019), 1-24.

Some 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.