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Structure of 2-step nilpotent ergodic averages for distinct-degree polynomials

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proves explicit L2 limits for multiple ergodic averages of totally ergodic 2-step nilpotent systems with distinct-degree polynomial iterates, and resolves the abelian joint ergodicity conjecture.

desk verdict Major paper that resolves the multidimensional joint ergodicity conjecture and gives the first explicit limits for 2-step nilpotent polynomial averages; the proofs are plausible but two gaps need filling before I'd trust the central theorem unconditionally. read the letter →

arxiv 2607.29368 v1 pith:7ISMYQKG submitted 2026-07-31 math.DS math.CO

Andreas Koutsogiannis , Borys Kuca , Wenbo Sun This is my paper · ORCID
classification math.DSmath.CO MSC 37A3011B3028D0537A44
keywords ergodicaverages2-stepnilpotentgroupactionsboxseminormsjointergodicitypolynomialiteratesnilsystemspopularcommondifferencesmultiplerecurrence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to describe, rather than only prove the existence of, the limit of multiple ergodic averages in which transformations T1,...,Tℓ generate a 2-step nilpotent group and the iterates are nonconstant polynomials of distinct degrees. The main theorem says that when each Tj is totally ergodic, the L2 limit of (1/N) Σ_{n≤N} f1(T1^{p1(n)}x)···fℓ(Tℓ^{pℓ(n)}x) equals the product of the integrals of f1,...,fℓ. Along the way the paper establishes box-seminorm estimates for such averages, derives popular-common-difference versions of the polynomial multiple recurrence theorem for 2-step nilpotent groups, and proves the joint ergodicity conjecture for multidimensional polynomials in Z^D-systems. It also gives a counterexample showing that the analogous joint ergodicity criterion fails in the 2-step nilpotent world, and it states that its methods do not extend beyond 2-step nilpotency or distinct-degree polynomials.

What carries the argument

The argument is carried by four interlocking components. First, a 2-step nilpotent version of the PET induction scheme: by repeatedly applying a standard difference (Cauchy-Schwarz-type) inequality, it eliminates the transformations T1,...,Tℓ-1 one by one, using a property called distinguishability to keep the Tℓ-coordinate independent of the commutator coordinates until only commuting transformations remain. Second, new abelian seminorm estimates (Theorem 1.18) control the resulting multiparameter averages by box seminorms attached to subgroups generated by all coefficient differences of the polynomials. Third, a comparison of box factors (Proposition 8.1) embeds each individual factor Z_s(

What would settle it

A concrete check is to take the 2-step nilpotent skew rotations on the two-dimensional torus given by T(x,y)=(x+a, y+2x+a) and S(x,y)=(x+b, y+2x+b) with 1,a,b rationally independent, and compute the L2 limit of (1/N)∑_{n=1}^N f1(T^n z) f2(S^{n^2} z) for continuous f1,f2. Theorem 1.5 predicts the product of the two integrals. A Fourier/spectral computation can settle this directly in the simplest noncommuting case. A failure would refute the theorem; a success would exercise the claimed mechanism and also test the factor inclusion Z_2(T) ⊆ Z_8(H).

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that 2-step nilpotent polynomial averages have a completely described limit: for totally ergodic T1,...,Tℓ and nonconstant polynomials p1,...,pℓ of distinct degrees, the average E_{n≤N} T1^{p1(n)}f1···Tℓ^{pℓ(n)}fℓ converges in L2 to ∫f1 dμ···∫fℓ dμ (Theorem 1.5). The accompanying seminorm estimates (Theorem 1.6) say the average tends to zero as soon as any one function has vanishing box seminorm of a suitable degree along its own transformation; this gives a characteristic factor for the average and supplies the technical engine for the paper's other results. The same machinery proves the full joint ergodicity conjecture for abelian Z^D-syst

Load-bearing premise

The load-bearing premise is Proposition 8.1: for a 2-step nilpotent system, every box factor of an individual generator T_j is contained in a sufficiently high-degree box factor of the whole group H; the proof of this inclusion uses a structural theorem for nilsystems (that measurable isomorphisms are affine) and does not obviously extend to higher-step groups. If this inclusion fails for some 2-step system, the explicit limit formula would not follow from the seminorm estima

Editorial extensions

If this is right

  • For every totally ergodic 2-step nilpotent system, any distinct-degree polynomial average has an explicit L2 limit equal to the product of the integrals.
  • The seminorm estimates make the characteristic factor explicit: if one function is orthogonal to the degree-s box factor of its transformation, the whole average vanishes.
  • In any finitely generated 2-step nilpotent group, every positive-density set contains syndetically many common differences of the form x, h1^{n^{d1}}x, ..., hℓ^{n^{dℓ}}x, with intersection density arbitrarily close to the natural density bound.
  • For Z^D-actions, the joint ergodicity conjecture is settled: product ergodicity plus difference ergodicity is necessary and sufficient for multidimensional polynomial sequences.
  • Totally ergodic nilrotations along independent polynomials produce jointly equidistributed orbits almost everywhere on nilsystems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the factor-inclusion exponent can be improved from O(s^2) to s, the approach might extend to higher-step nilpotent groups; the quadratic overhead looks like an artifact of the comparison argument, and the paper's open problems highlight this as a natural target.
  • The counterexample to the naive nilpotent joint ergodicity criterion suggests that joint ergodicity for nilpotent actions will need additional conditions on the action of the commutator subgroup, not just product and difference ergodicity.
  • The new abelian seminorm estimates, being uniform across functions and expressed in terms of full coefficient subgroups, may be the right input for quantitative finite-N bounds; the paper itself notes that quantification is open.
  • The distinguishability trick for keeping Tℓ separate from commutators is likely the key noncommutative innovation; testing it on the model average T^n S^{n^2} for a 3-step nilpotent pair would reveal whether the method can go further.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper develops seminorm estimates and limiting formulas for multiple ergodic averages of the form N^{-1}\sum_{n=1}^N T_1^{p_1(n)}f_1\cdots T_\ell^{p_\ell(n)}f_\ell for measure-preserving actions generated by 2-step nilpotent groups and nonconstant polynomials of distinct degrees. The central results are Theorem 1.5 (total ergodicity implies L^2 convergence to the product of integrals), Theorem 1.6 (Host-Kra seminorm estimates), Theorem 1.8/Corollary 1.9 (popular common differences), Theorem 1.13 (full resolution of the joint ergodicity conjecture for polynomial Z^D-actions), Theorem 1.15 (a joint ergodicity criterion in the 2-step nilpotent setting), Theorems 1.14 and 1.16 (counterexamples to naive nilpotent analogues), and Theorem 1.18 (new abelian box-seminorm estimates). The proof combines a 2-step nilpotent PET induction, relative concatenation and seminorm smoothing, Host-Kra factor comparisons via affine-isomorphism theorems, and equidistribution results on nilsystems.

Significance. If the proofs are correct, this is a substantial advance: it gives the first explicit limiting formulas for genuinely noncommutative polynomial multiple averages, closes the abelian joint ergodicity conjecture, and supplies new seminorm control that is likely to be useful beyond the present applications. The paper is largely proof-heavy and self-contained, and the model computation in §6 convincingly illustrates the PET mechanism. The counterexamples to the nilpotent analogue of joint ergodicity are valuable and clearly presented. The main caveats are a load-bearing gap in the use of Parry's theorem in §8.1 and an explicitly omitted variant of Proposition 9.3 that is invoked in the discussion around Theorem 1.15; both need to be resolved before the central claims can be regarded as fully proven.

major comments (3)
  1. [§8.1, Proposition 8.2] The proof uses Parry's theorem to conclude that S_{\ell+1} is affine, but the hypotheses of Parry's theorem are not verified. The conjugated transformations \hat S_j = S_{\ell+1}S_jS_{\ell+1}^{-1} are not shown to be nilrotations on the original nilmanifold G/\Gamma, so the target system (Y,\hat S_1,\dots,\hat S_\ell) is not known to be a nilsystem before applying Parry. As written, the proof even contains an apparent typo: the conjugating map is called S_\ell, not S_{\ell+1}. This is not a cosmetic issue: Proposition 8.2 is the bridge that lets the authors identify Z_{s'}(H) factors and pass from the seminorm estimates of Theorem 1.6 to the limiting formula of Theorem 1.5. The authors should either prove a normalizer-type theorem for ergodic nilsystems or give an independent argument that S_{\ell+1} is affine with respect to the original nilmanifold structure.
  2. [§9.1, after Proposition 9.3] The text explicitly states that Proposition 9.3 does not extend to integer-valued polynomials and that 'a more complicated version' sufficient for Theorem 1.15 is omitted. This is a flagged missing proof. If Theorem 1.15 is intended only for p_j\in Z[n], the omission is not load-bearing and the remark should be clarified. But if the theorem or its applications cover integer-valued polynomials, the proof of Theorem 1.15 is incomplete as it stands. The manuscript should either include the promised variant or explicitly restrict the statement and remove the implication that the omitted result is needed.
  3. [§10, proof of Theorem 1.13] The final reduction from the box-seminorm control in Theorem 1.18 to control by a single Host-Kra seminorm |||f_j|||_{s,T_j} is compressed. In a general Z^D-system the notation T_j is not defined, and the passage through 'only ergodic subgroups' followed by (26) hides several nontrivial steps: one must justify why the subgroups H_{j,j'} can be replaced by a single ergodic subgroup and why that subgroup gives the same seminorm as the one generated by the relevant coordinate action. This is a central step in resolving Conjecture 1.12, so the argument should be written out or the relevant known lemma should be quoted precisely.
minor comments (3)
  1. [§6, Section 2 overview] Section 2.1 says that Section 6 is dedicated to the proof of 'Theorem 1.18 for the model average', but Theorem 6.1 is a special case of Theorem 1.6, not of Theorem 1.18. Please correct the cross-reference.
  2. [§8.1, Proposition 8.2] The proof has several apparent typos: the isomorphism map should be S_{\ell+1}, not S_\ell, in the first paragraph, and later 'S_\ell agrees m_Y-a.e. with an affine map' should presumably refer to S_{\ell+1}. These typos make the already delicate argument harder to verify.
  3. [§1.2, footnote 4] The footnote contains a duplicated word: 'applies applies'. Please correct.

Circularity Check

0 steps flagged · score 1.0 of 10

No definitional circularity; only routine citations of prior (including co-authored) theorems, none of which are the target result.

full rationale

I walked the derivation chain: Theorem 1.5 rests on Theorem 1.6, Proposition 8.1, the Candela–Szegedy structure theorem (Theorem 1.4), and the equidistribution result Theorem 1.20. Theorem 1.6 rests on the local PET and Theorem 1.18. Theorem 1.13 rests on Theorem 1.18 plus external joint-ergodicity criteria. The only places where the paper invokes work by the same authors are Theorem 3.15 (the relative concatenation theorem from [19]) and Theorem 1.17 (existing seminorm estimates from [18,20,57]). These are cited as previously proved theorems with independent content; neither is the statement being derived nor a consequence of it. The paper even flags the dependence: 'We do emphasize, though, that the proof of Theorem 1.18 is built upon Theorem 1.17.' No parameter is fitted to data and no claimed limit is defined in terms of the average it is supposed to describe. The skeptical concern about Proposition 8.2 and Parry's affine-isomorphism theorem is a correctness risk about a missing verification, not a reduction of the result to its own input. Hence no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claims rest on published deep theorems and standard ergodic theory tools rather than fitted constants or newly invented objects. No free parameters are introduced. The paper's own new structures (distinguishability, nice families, pinned families) are definitions used internally, not entities with independent empirical handles.

assumptions (8)
  • standard math Candela-Szegedy structure theorem for nilpotent Host-Kra factors (Theorem 1.4): for ergodic H-actions, the Host-Kra factor is an s-step H-pronilsystem.
    Invoked in proofs of Theorem 1.5 and Proposition 8.1; the paper argues its cubic measures match the Candela-Szegedy definitions.
  • standard math Walsh's nilpotent norm convergence theorem (Theorem 1.3): polynomial multiple averages converge in L^2 for nilpotent systems.
    Used in Sections 6 and 7 to replace iterated limits by single multiparameter limits.
  • standard math Bergelson-Leibman Fubini-type principle for averages [8, Lemma 1.1].
    Used to justify swapping and merging averages in the van der Corput arguments.
  • standard math Parry's theorem that measurable isomorphisms of nilsystems are affine, as cited in [69] and [70, Theorem 1.4].
    Basis of Proposition 8.2 and Corollary 8.3, which feed into Proposition 8.1.
  • standard math Leibman's equidistribution theorem for polynomial orbits on nilmanifolds [64, 65].
    Used in Theorem 1.20 to reduce equidistribution on nilsystems to the torus case.
  • standard math Relative concatenation theorem [19, Theorem 3.1], quoted as Theorem 3.15.
    Core tool in Proposition 5.7 for removing seminorm subgroups.
  • standard math Joint ergodicity criteria for Z^D-actions [3, Theorem 1.1].
    Used in the proof of Theorem 1.13 to convert seminorm control plus good equidistribution into joint ergodicity.
  • standard math Mean ergodic theorem for amenable group actions and Gowers-Cauchy-Schwarz inequalities for box seminorms.
    Foundational for defining cubic measures and box seminorms in Section 3.

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Cite this review

Pith. "Pith review of Structure of 2-step nilpotent ergodic averages for distinct-degree polynomials." pith.science (2026). https://pith.science/paper/7ISMYQKG

@misc{pith2026260729368,
  author       = {Pith},
  title        = {Pith review of: Structure of 2-step nilpotent ergodic averages for distinct-degree polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ISMYQKG}},
  note         = {Machine review of arXiv:2607.29368}
}
abstract

We investigate manifestations of the Nilpotent Heuristic, which posits that recurrence and convergence phenomena known for measure-preserving $\mathbb{Z}^D$-systems extend to nilpotent group actions. Our main results establish seminorm estimates and limiting formulas for multiple ergodic averages arising from actions of 2-step nilpotent groups. In particular, if $T_1,\ldots,T_\ell$ are totally ergodic and generate a 2-step nilpotent group, then \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T_1^n f_1 \cdots T_\ell^{n^\ell}f_\ell = \prod_{j=1}^{\ell}\int f_j\,d\mu \] in the $L^{2}$ norm for all bounded functions $f_{1},\dots,f_{\ell}$; the same holds for any distinct-degree polynomial iterates. We also obtain popular-common-difference versions of the polynomial Szeme\'edi theorem in the same setting. In a different direction, our approach allows us to completely resolve the joint ergodicity conjecture for multidimensional polynomials and $\mathbb Z^D$-systems; we also present an example showing that, surprisingly enough, the 2-step nilpotent analog fails. We conclude with many open problems concerning joint ergodicity, seminorm estimates, and the structure theory of nilpotent systems.

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Reference graph

Works this paper leans on

85 extracted references · 5 linked inside Pith

  1. [1]

    T. Austin. Ajtai–Szemerédi theorems over quasirandom groups. InRecent Trends in Combinatorics, pages 453–484. Springer, 2016

    2016

  2. [2]

    Berend and V

    D. Berend and V. Bergelson. Jointly ergodic measure-preserving transformations.Israel J. Math., 49(4):307–314, 1984

    1984

  3. [3]

    Bergelson and A

    V. Bergelson and A. Ferré-Moragues. An ergodic correspondence principle, invariant means and applications.Israel J. Math.,245(2):921–962, 2021

    2021

  4. [4]

    Bergelson and A

    V. Bergelson and A. Leibman. Polynomial extensions of van der Waerden’s and Szemerédi’s theo- rems.J. Amer. Math. Soc.,9:725–753, 1996

    1996

  5. [5]

    Bergelson and A

    V. Bergelson and A. Leibman. A nilpotent Roth theorem.Invent. Math.,147:429–470, 2002

    2002

  6. [6]

    Bergelson and A

    V. Bergelson and A. Leibman. Topological multiple recurrence for polynomial configurations in nilpotent groups.Adv. Math.,175(2):271–296, 2003

    2003

  7. [7]

    Bergelson and A

    V. Bergelson and A. Leibman. Failure of the Roth theorem for solvable groups of exponential growth. Ergodic Theory Dynam. Systems,24(1):45–53, 2004

    2004

  8. [8]

    Bergelson and A

    V. Bergelson and A. Leibman. Cubic averages and large intersections. InRecent Trends in Ergodic Theory and Dynamical Systems, volume631, pages 5–19. Contemporary Mathematics, 2015

    2015

Show all 85 references
  1. [9]

    Bergelson, A

    V. Bergelson, A. Leibman, and Y. Son. Joint ergodicity along generalized linear functions.Ergodic Theory Dynam. Systems,36(7):2044–2075, 2016

  2. [10]

    Bergelson and Y

    V. Bergelson and Y. Son. Joint ergodicity of piecewise monotone interval maps.Nonlinearity, 36:3376–3418, 2023

    2023

  3. [11]

    Best and A

    A. Best and A. Ferré-Moragues. Polynomial ergodic averages for certain countable ring actions. Discrete Contin. Dyn. Syst.,42(7):3379–3413, 2022

    2022

  4. [12]

    P. Candela. Notes on compact nilspaces.Discrete Anal., 09 2017

    2017

  5. [13]

    P. Candela. Notes on nilspaces: algebraic aspects.Discrete Anal., 09 2017

    2017

  6. [14]

    Systems,40(11):3015–3029, 2020

    P.Candela, D.González-Sánchez, andB.Szegedy.Onnilspacesystemsandtheirmorphisms.Ergodic Theory Dynam. Systems,40(11):3015–3029, 2020

    2020

  7. [15]

    Candela and B

    P. Candela and B. Szegedy.Nilspace factors for general uniformity seminorms, cubic exchangeability and limits, volume287ofMem. Amer. Math. Soc.Amer. Math. Soc., 2023

    2023

  8. [16]

    Christ, P

    M. Christ, P. Durcik, and J. Roos. Trilinear smoothing inequalities and a variant of the triangular Hilbert transform.Adv. Math.,390:107863, 2021

    2021

  9. [17]

    Q. Chu, N. Frantzikinakis, and B. Host. Ergodic averages of commuting transformations with dis- tinct degree polynomial iterates.Proc. Lond. Math. Soc.,102:801–842, 2011

    2011

  10. [18]

    Donoso, A

    S. Donoso, A. Ferré-Moragues, A. Koutsogiannis, and W. Sun. Decomposition of multicorrelation sequences and joint ergodicity.Ergodic Theory Dynam. Systems,44(2):432–480, 2024

    2024

  11. [19]

    Donoso, A

    S. Donoso, A. Koutsogiannis, B. Kuca, W. Sun, and K. Tsinas. Resolving the joint ergodicity problem for Hardy sequences. Preprint 2025, arXiv:2506.20459

    2025 arXiv

  12. [20]

    Donoso, A

    S. Donoso, A. Koutsogiannis, B. Kuca, W. Sun, and K. Tsinas. Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences. Preprint 2024, arXiv:2410.15130

    2024

  13. [21]

    Donoso, A

    S. Donoso, A. Koutsogiannis, and W. Sun. Seminorms for multiple averages along polynomials and applications to joint ergodicity.J. Anal. Math.,146(1):1–64, 2022

    2022

  14. [22]

    Donoso, A

    S. Donoso, A. Koutsogiannis, and W. Sun. Joint ergodicity for functions of polynomial growth. Israel J. Math., pages 1–49, 2025

    2025

  15. [23]

    Donoso, A

    S. Donoso, A. Koutsogiannis, and W. Sun. Joint transitivity for linear iterates.Forum Math. Sigma, 13:e34, 2025

    2025

  16. [24]

    Donoso and W

    S. Donoso and W. Sun. Quantitative multiple recurrence for two and three transformations.Israel J. Math.,226(1):71–85, 2018

    2018

  17. [25]

    Frantzikinakis

    N. Frantzikinakis. Equidistribution of sparse sequences on nilmanifolds.J. Anal. Math.,109(1):353– 395, 2009

    2009

  18. [26]

    Frantzikinakis

    N. Frantzikinakis. Multiple recurrence and convergence for Hardy sequences of polynomial growth. J. Anal. Math.,112(1):79–135, 2010

    2010

  19. [27]

    Frantzikinakis

    N. Frantzikinakis. A multidimensional Szemerédi theorem for Hardy sequences of different growth. Trans. Amer. Math. Soc.,367:5653–5692, 2012

    2012

  20. [28]

    Frantzikinakis

    N. Frantzikinakis. Joint ergodicity of sequences.Adv. Math.,417:108918, 2023

    2023

  21. [29]

    Frantzikinakis, Host B., and Kra B

    N. Frantzikinakis, Host B., and Kra B. The polynomial multidimensional szemerédi theorem along shifted primes.Israel J. Math.,194:331–348, 2010

    2010

  22. [30]

    Frantzikinakis and B

    N. Frantzikinakis and B. Host. Higher order Fourier analysis of multiplicative functions and appli- cations.J. Amer. Math. Soc.,30(1):67–157, 2017

    2017

  23. [31]

    Frantzikinakis and B

    N. Frantzikinakis and B. Host. Weighted multiple ergodic averages and correlation sequences.Er- godic Theory Dynam. Systems,38(1):81–142, 2018

    2018

  24. [32]

    Frantzikinakis, O

    N. Frantzikinakis, O. Klurman, and J. Moreira. Partition regularity of Pythagorean pairs.Forum Math. Pi,13:e5, 2025. 74

    2025

  25. [33]

    Frantzikinakis and B

    N. Frantzikinakis and B. Kra. Polynomial averages converge to the product of integrals.Israel J. Math.,148(1):267–276, 2005

    2005

  26. [34]

    Frantzikinakis and B

    N. Frantzikinakis and B. Kra. Ergodic averages for independent polynomials and applications.J. Lond. Math. Soc.,74:131–142, 2006

    2006

  27. [35]

    Frantzikinakis and B

    N. Frantzikinakis and B. Kuca. Seminorm control for ergodic averages with commuting transforma- tions along pairwise dependent polynomials.Ergodic Theory Dynam. Systems,43(12):4074–4137, 2023

    2023

  28. [36]

    Frantzikinakis and B

    N. Frantzikinakis and B. Kuca. Joint ergodicity for commuting transformations and applications to polynomial sequences.Invent. Math.,239(2):621–706, 2025

    2025

  29. [37]

    Furstenberg

    H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.J. Anal. Math.,31(1):204–256, 1977

    1977

  30. [38]

    Furstenberg.Recurrence in Ergodic Theory and Combinatorial Number Theory

    H. Furstenberg.Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Uni- versity Press, 1981

    1981

  31. [39]

    Furstenberg and Y

    H. Furstenberg and Y. Katznelson. An ergodic Szemerédi theorem for commuting transformations. J. Anal. Math.,34(1):275–291, 1978

    1978

  32. [40]

    Glasner.Ergodic theory via joinings

    E. Glasner.Ergodic theory via joinings. Mathematical surveys and monographs. Amer. Math. Soc., 2003

    2003

  33. [41]

    W. T. Gowers. Quasirandom groups.Combin. Probab. Comput.,17(3):363–387, 2008

    2008

  34. [42]

    Green and T

    B. Green and T. Tao. The primes contain arbitrarily long arithmetic progressions.Ann. of Math., 167(2):481–547, 2008

    2008

  35. [43]

    Green and T

    B. Green and T. Tao. Linear equations in primes.Ann. of Math.,171:1753–1850, 2010

    2010

  36. [44]

    Green and T

    B. Green and T. Tao. The Möbius function is strongly orthogonal to nilsequences.Ann. of Math., 175(2):541–566, 2012

    2012

  37. [45]

    B. Host. Ergodic seminorms for commuting transformations and applications.Studia Math.,195:31– 49, 2009

    2009

  38. [46]

    Host and B

    B. Host and B. Kra. Convergence of polynomial ergodic averages.Israel J. Math.,149(1):1–19, 2005

    2005

  39. [47]

    Host and B

    B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds.Ann. of Math.,161(1):397– 488, 2005

    2005

  40. [48]

    Host and B

    B. Host and B. Kra.Nilpotent structures in ergodic theory. AMS, 2018

    2018

  41. [49]

    Hu and V

    B. Hu and V. Lie. On the curved trilinear Hilbert transform. Preprint 2023, arXiv:2308.10706

    2023 arXiv

  42. [50]

    Huang, S

    W. Huang, S. Shao, and X. Ye. Topological correspondence of multiple ergodic averages of nilpotent group actions.J. Anal. Math.,138(2):687–715, 2019

    2019

  43. [51]

    A. D. Ionescu, Á. Magyar, M. Mirek, and T. Z. Szarek. Polynomial averages and pointwise ergodic theorems on nilpotent groups.Invent. Math.,231(3):1023–1140, 2023

    2023

  44. [52]

    D. Kosz, M. Mirek, S. Peluse, R. Wan, and J. Wright. The multilinear circle method and a question of Bergelson.Ann. of Math.,203(1):233–359, 2026

    2026

  45. [53]

    Koutsogiannis and W

    A. Koutsogiannis and W. Sun. Total joint ergodicity for totally ergodic systems. Preprint 2023, arXiv:2302.12278 to appear inIsrael J. Math

    2023 arXiv

  46. [54]

    B. Kra, J. Moreira, F. K. Richter, and D. Robertson. The density finite sums theorem.Invent. Math.,243(1):1–31, 2026

    2026

  47. [55]

    Krause, M

    B. Krause, M. Mirek, and T. Tao. Pointwise ergodic theorems for non-conventional bilinear poly- nomial averages.Ann. of Math.,195:997–1109, 2022

    2022

  48. [56]

    Krause, H

    B. Krause, H. Mousavi, T. Tao, and J. Teräväinen. Pointwise convergence of bilinear polynomial averages over the primes.Ergodic Theory Dynam. Systems, 45:3760–3799, 2025

    2025

  49. [57]

    Kravitz, B

    N. Kravitz, B. Kuca, and J. Leng. Quantitative concatenation for polynomial box norms.Adv. Math.,489:110820, 2026

    2026

  50. [58]

    B. Kuca. Joint ergodicity - 40 years on. Preprint 2026, arXiv:2603.18974

    2026

  51. [59]

    B. Kuca. Multidimensional polynomial patterns over finite fields: bounds, counting estimates and Gowers norm control.Adv. Math.,448:109700, 2024

    2024

  52. [60]

    Lacey and C

    M. Lacey and C. Thiele.Lp estimates on the bilinear Hilbert transform for2< p <∞.Ann. of Math.,146(3):693–724, 1997

    1997

  53. [61]

    A. Leibman. Multiple recurrence theorem for measure preserving actions of a nilpotent group.Geom. Funct. Anal.,8(5):853–931, 1998

    1998

  54. [62]

    A. Leibman. Polynomial mappings of groups.Israel J. Math.,129:29–60, 2002

    2002

  55. [63]

    A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables.Israel J. Math.,146:303–315, 2005

    2005

  56. [64]

    A. Leibman. Pointwise convergence of ergodic averages for polynomial actions ofZd by translations on a nilmanifold.Ergodic Theory Dynam. Systems,25(1):215–225, 2005

    2005

  57. [65]

    A. Leibman. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold.Ergodic Theory Dynam. Systems,25(1):201–213, 2005. 75

    2005

  58. [66]

    M. Mirek. The circle method and pointwise ergodic theorems. InProc. ICM 2026, volume4, pages 698–718

    2026

  59. [67]

    Mirek, R

    M. Mirek, R. Wan, and J. Wright. Pointwise convergence of polynomial multiple ergodic averages along the primes. Preprint 2025, arXiv:2505.15549

    2025 arXiv

  60. [68]

    J. Moreira. Monochromatic sums and products inN.Ann. of Math.,185(3):1069–1090, 2017

    2017

  61. [69]

    W. Parry. Metric classification of ergodic nilflows and unipotent affines.Amer. J. Math.,93(3):819– 828, 1971

    1971

  62. [70]

    W. Parry. Dynamical representations in nilmanifolds.Compos. Math.,26(2):159–174, 1973

    1973

  63. [71]

    S. Peluse. Mixing for three-term progressions in finite simple groups.Math. Proc. Cambridge Philos. Soc,165(2):279–286, 2018

    2018

  64. [72]

    S. Peluse. On the polynomial Szemerédi theorem in finite fields.Duke Math. J.,168(5):749–774, 2019

    2019

  65. [73]

    Peluse and S

    S. Peluse and S. Prendiville. Quantitative bounds in the non-linear Roth theorem.Invent. Math., 238(3):865–903, 2024

    2024

  66. [74]

    J. Qiu. Polynomial orbits in totally minimal systems.Adv. Math.,432:109260, 2023

    2023

  67. [75]

    F. Richter. Uniform distribution in nilmanifolds along functions from a Hardy field.J. Anal. Math., 149:421–483, 2023

    2023

  68. [76]

    T. Tao. Poincarés legacies: pages from year two of a mathematical blog.https://terrytao. wordpress.com/wp-content/uploads/2009/01/whatsnew.pdf

    2009

  69. [77]

    T. Tao. Mixing for progressions in nonabelian groups.Forum Math. Sigma,1:e2, 2013

    2013

  70. [78]

    T. Tao. Cancellation for the multilinear Hilbert transform.Collect. Math.,67:191–206, 2016

    2016

  71. [79]

    Tao and T

    T. Tao and T. Ziegler. The primes contain arbitrarily long polynomial progressions.Acta Math., 201(2):213 – 305, 2008

    2008

  72. [80]

    Tao and T

    T. Tao and T. Ziegler. Concatenation theorems for anti-Gowers uniform functions and Host-Kra characteristic factors.Discrete Anal.,13, 2016

    2016

  73. [81]

    Teräväinen

    J. Teräväinen. Pointwise convergence of ergodic averages with Möbius weight. Preprint 2024, arXiv:2401.03174

    2024 arXiv

  74. [82]

    K. Tsinas. Pointwise convergence in nilmanifolds along smooth functions of polynomial growth. Ergodic Theory Dynam. Systems,44(7):1963–2008, 2024

    1963

  75. [83]

    M. Walsh. Norm convergence of nilpotent ergodic averages.Ann. of Math.,175(3):1667–1688, 2012

    2012

  76. [84]

    Zorin-Kranich

    P. Zorin-Kranich. A nilpotent IP polynomial multiple recurrence theorem.J. Anal. Math., 123(1):183–225, 2014

    2014

  77. [85]

    Zorin-Kranich

    P. Zorin-Kranich. Norm convergence of multiple ergodic averages on amenable groups.J. Anal. Math.,130(1):219–241, 2016. (Andreas Koutsogiannis)Department of Mathematics, Aristotle University of Thessa- loniki, Thessaloniki 54124, Greece Email address:akoutsogiannis@math.auth.g...

    2016

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