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Some open problems on multiple ergodic averages (version of 2016)

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Progress since 2016 (Latest update: January 2023.)

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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*.
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***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.
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***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.
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***Problem 10.**
Some progress made by A. Koutsogiannis.

Multiple ergodic averages for variable polynomials.

*Discrete and Continuous Dynamical Systems*,
**42**, (2022), 4637-4668.

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***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.
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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.
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***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.
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and for the third sequence in

Interpolation sets and nilsequences.

*Colloq. Math.*,
**162**, (2020), 181-199.
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***Problem 15.** **Solved** by N. F. and B. Kuca.

Joint ergodicity for commuting transformations and applications to polynomial sequences.

Preprint.

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***Problem 16.** **Solved** by N. F. and B. Kuca.

Joint ergodicity for commuting transformations and applications to polynomial sequences.

Preprint.

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***Problem 17.** **Solved** by N. F. and B. Kuca.

Joint ergodicity for commuting transformations and applications to polynomial sequences.

Preprint.

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***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.
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***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.
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***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*.

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***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.

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***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.

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***Problem 31.**
Some progress made by J. Briet, Z. Dvir, S. Gopi.
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Outlaw distributions and locally decodable codes.

*Theory of Computing*.
** 15**, (2019), 1-24.
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Some progress made by J. Briet, S. Gopi.
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Gaussian width bounds with applications to arithmetic progressions in random settings.

*International Mathematics Research Notices*.
** 2020**, no. 22, (2020), 8673-8696.
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