# Nilsystem

A nilsystem is a dynamical system whose state space is a compact nilmanifold \( G/\Gamma \), where \( G \) is a nilpotent [Lie group](https://www.edgepedia.org/lie-group) and \( \Gamma \) a discrete cocompact subgroup, and whose dynamics is translation by a fixed element of \( G \).<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup> Nilsystems generalize rotations on tori to nonabelian nilpotent groups and are the structured objects that describe the behavior of multiple ergodic (time averages equal space averages; orbits spread uniformly) averages.<sup>[2](https://bookstore.ams.org/surv-236)</sup> They sit at the intersection of ergodic theory, topological dynamics, additive combinatorics, and number theory.<sup>[2](https://bookstore.ams.org/surv-236)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( (X, T) \) with \( X = G/\Gamma \) an \( s \)-step nilmanifold and \( T \) the translation by a fixed \( \tau \in G \); Haar measure makes it measure-preserving<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup><sup> • </sup><sup>[3](https://ar5iv.labs.arxiv.org/html/2307.07213)</sup> |
| Ergodicity equivalences | Transitivity, minimality, unique ergodicity, and ergodicity with respect to Haar measure are all equivalent<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup> |
| Characteristic factors | The factor characteristic for \( k \)-term multiple ergodic averages is a \( (k-1) \)-step pro-nilsystem (Host–Kra, Ziegler)<sup>[4](https://people.math.osu.edu/leibman.1/papers/hkz.pdf)</sup><sup> • </sup><sup>[5](https://www.ams.org/journals/jams/2007-20-01/S0894-0347-06-00532-7/)</sup> |
| Gowers norms | The \( k \)-th Gowers–Host–Kra seminorm corresponds to a \( (k-1) \)-step nilsystem<sup>[6](https://sites.math.northwestern.edu/~kra/papers/nil.pdf)</sup> |
| Inverse theorem | A function with large \( U^{s+1}[N] \) norm correlates with a bounded-complexity \( s \)-step nilsequence, with bounds depending only on \( s \) and the error \( \delta \)<sup>[7](https://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p11-p.pdf)</sup> |
| Quantitative equidistribution | Polynomial orbits on nilmanifolds factor as \( g = \epsilon g_0 \gamma \) with bounds uniform in \( N \) and polynomial in \( \delta^{-1} \) (Green–Tao 2012)<sup>[8](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p02-p.pdf)</sup> |
| Mixing behavior | Ergodic \( k \)-step nilsystems with \( k \ge 2 \) have compact-Lebesgue spectrum<sup>[3](https://ar5iv.labs.arxiv.org/html/2307.07213)</sup> |

## Definition and first examples

Let \( s \ge 2 \) be an integer, \( G \) an \( s \)-step nilpotent Lie group, and \( \Gamma \) a discrete cocompact subgroup. The compact manifold \( X = G/\Gamma \) is an \( s \)-step nilmanifold, and the pair \( (X, T) \), where \( T \) is translation by a fixed \( \tau \in G \), is an \( s \)-step nilsystem, both as a topological system and, with [Haar measure](https://www.edgepedia.org/haar-measure), as a measure-preserving system.<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup> Every nilmanifold carries a unique left-translation-invariant Borel probability measure, the Haar measure, so each topological nilsystem is automatically a measure-preserving system.<sup>[3](https://ar5iv.labs.arxiv.org/html/2307.07213)</sup> In coordinates, \( T_\tau(g\Gamma) = \tau g \Gamma \) is left translation.<sup>[9](https://arxiv.org/html/2602.21021)</sup>

**The abelian case is a torus rotation.** Nilsystems are a natural generalization of rotations and arise in the study of multiple ergodic averages, the structural analysis of topological dynamical systems, and asymptotics for patterns in subsets of the integers.<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup>

**Concrete nonabelian examples.** The Heisenberg nilsystem is built from the [Heisenberg group](https://www.edgepedia.org/heisenberg-group) \( H_d \), whose commutator subgroup lies in the center, endowed with translation by an element \( \tau \).<sup>[10](https://www.numdam.org/article/AIF_2008__58_5_1407_0.pdf)</sup> The skew shift is a circle extension of a circle shift, and the Heisenberg nilsystem is a circle extension of an abelian 2-torus shift.<sup>[11](https://terrytao.wordpress.com/2008/03/09/254a-lecture-16-a-ratner-type-theorem-for-nilmanifolds/)</sup>

**Nilsequences.** A basic \( r \)-step nilsequence is a sequence \( \psi(n) = f(\eta(n)x) \), where \( x \) is a point of an \( r \)-step nilmanifold \( X = G/\Gamma \), \( \eta : \mathbb{Z}^d \to G \) is a homomorphism, and \( f \in C(X) \).<sup>[12](https://people.math.osu.edu/leibman.1/papers/msqx.pdf)</sup> In the two-step case, a 2-step nilsequence is a uniform limit of basic 2-step nilsequences.<sup>[10](https://www.numdam.org/article/AIF_2008__58_5_1407_0.pdf)</sup>

## Nilmanifolds, nilpotence, and step structure

The step of a nilsystem measures how far \( G \) is from abelian. The step controls the system's complexity in every direction: topological nilsystems have complexity of polynomial growth, with an explicit-degree polynomial serving as an asymptotic bound both from below and above.<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup>

**Inductive structure.** Every \( s \)-step nilsystem with \( G \) connected and simply connected is built inductively as a toral extension: it is a torus extension of an \( (s-1) \)-step nilsystem.<sup>[11](https://terrytao.wordpress.com/2008/03/09/254a-lecture-16-a-ratner-type-theorem-for-nilmanifolds/)</sup> This layering is what makes nilsystems tractable: properties can be proved at the torus level and lifted one commutator stage at a time. A related reduction holds for minimal nilsystems: writing \( G_0 \) for the subgroup spanned by the identity component of \( G \) and \( \tau \), and \( \Gamma_0 = \Gamma \cap G_0 \), one has \( G = G_0 \Gamma \) and \( (X, T) \cong (G_0/\Gamma_0, T_0) \).<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup>

## Dynamical properties

Nilsystems are unusually rigid. For nilsystems, transitivity, minimality, unique ergodicity, and ergodicity with respect to Haar measure are all equivalent.<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup> Consequently every ergodic nilsystem is uniquely ergodic, so every orbit is equidistributed with respect to Haar measure.<sup>[11](https://terrytao.wordpress.com/2008/03/09/254a-lecture-16-a-ratner-type-theorem-for-nilmanifolds/)</sup>

**Distality and orbit closures.** Nilsystems are distal, so any two orbit closures either coincide or are disjoint.<sup>[13](https://pure.uva.nl/ws/files/1693302/126537_08.pdf)</sup> A Ratner-type theorem sharpens this: for a nilsystem with \( G \) connected and simply connected, an orbit partitions into finitely many congruence classes, on each of which it is equidistributed in a subnilmanifold \( Y = Hx \) for a closed connected subgroup \( H \) of \( G \).<sup>[11](https://terrytao.wordpress.com/2008/03/09/254a-lecture-16-a-ratner-type-theorem-for-nilmanifolds/)</sup>

**Spectrum and mixing.** For \( k \ge 2 \), an ergodic \( k \)-step nilsystem has compact-Lebesgue spectrum on \( L^2 \) of the nilmanifold, which settles the maximal spectral type problem for nilsystems.<sup>[3](https://ar5iv.labs.arxiv.org/html/2307.07213)</sup>

## Characteristic factors: the Host–Kra–Ziegler theory

The central role of nilsystems in ergodic theory comes from multiple recurrence. After Conze and Lesigne showed that the characteristic factor for three-term multiple ergodic averages is an inverse limit of two-step nilsystems, it was natural to conjecture that the characteristic factor for \( k \)-term averages is an inverse limit of \( (k-1) \)-step nilsystems; Host, Kra, and Ziegler confirmed this.<sup>[4](https://people.math.osu.edu/leibman.1/papers/hkz.pdf)</sup>

The factors are defined inductively. The Kronecker factor \( Z_1(X) \) is the factor spanned by the eigenfunctions of the system, isomorphic to a translation on a compact abelian group with Haar measure; the factor \( Z_s(X) \) is the inverse limit of all \( s \)-step nilsystems that are factors of \( (X, \mathcal{B}, \mu, T) \), and all \( Z_s \) are trivial for weakly mixing systems.<sup>[1](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)</sup> Ziegler proved that there exists a unique \( k \)-universal characteristic factor for averaging schemes of length \( k \), and that it has the structure of a \( (k-1) \)-step pro-nilsystem, more specifically an inverse limit of \( (k-1) \)-step nilflows; her construction extends Conze–Lesigne's by building \( Y_k \) as a skew-shift product \( Y_{k-1} \times H \).<sup>[5](https://www.ams.org/journals/jams/2007-20-01/S0894-0347-06-00532-7/)</sup><sup> • </sup><sup>[4](https://people.math.osu.edu/leibman.1/papers/hkz.pdf)</sup>

**Convergence of multiple averages.** Using this factor, Ziegler showed that the averages \( \frac{1}{N} \sum_{n=1}^{N} \prod_{j=1}^{k} f_j(T^{a_j n}x) \), for distinct nonzero integers \( a_j \), converge in \( L^2(\mu) \), providing an alternative proof to Host–Kra's.<sup>[5](https://www.ams.org/journals/jams/2007-20-01/S0894-0347-06-00532-7/)</sup> By the structure theorem, systems of order \( k \) correspond to pro-nilsystems, where such a \( k \)-step pro-nilsystem is obtained as the inverse limit, within the category of measure-preserving systems, of \( k \)-step nilsystems, each finite stage carrying Haar measure and the connecting maps intertwining the nilrotations.<sup>[13](https://pure.uva.nl/ws/files/1693302/126537_08.pdf)</sup><sup> • </sup><sup>[14](https://arxiv.org/html/2608.22508)</sup>

## Nilsystems in higher-order Fourier analysis and the Gowers norms

Gowers introduced his norms for functions on \( \mathbb{Z}/N\mathbb{Z} \) in his proof of [Szemerédi's theorem](https://www.edgepedia.org/szemeredis-theorem), and Green and Tao adapted them to show that the primes contain arbitrarily long arithmetic progressions.<sup>[6](https://sites.math.northwestern.edu/~kra/papers/nil.pdf)</sup> In ergodic theory, the third Gowers seminorm corresponds to a 2-step nilsystem, and more generally the \( k \)-th seminorm corresponds to a \( (k-1) \)-step nilsystem; the second seminorm is characterized completely by the Kronecker factor, the largest abelian group rotation factor.<sup>[6](https://sites.math.northwestern.edu/~kra/papers/nil.pdf)</sup>

**The inverse theorem.** The Green–Tao–Ziegler inverse theorem for the Gowers \( U^{s+1}[N] \)-norm states that if \( f : [N] \to [-1, 1] \) satisfies \( \|f\|_{U^{s+1}[N]} > \delta \), then \( f \) correlates with a bounded-complexity \( s \)-step nilsequence \( F(g(n)\Gamma) \), with the bounds on complexity and correlation depending only on \( s \) and \( \delta \).<sup>[7](https://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p11-p.pdf)</sup> Combined with previous results, the inverse conjecture implies the Hardy–Littlewood prime tuples conjecture for any linear system of finite complexity.<sup>[7](https://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p11-p.pdf)</sup>

**Nilspace alternative.** Szegedy's approach to the inverse theorem uses nilspaces, an abstract combinatorial analog of nilmanifolds, as an alternative to the Green–Tao–Ziegler approach; nilspaces were also used by Gutman, Manners, and Varjú to generalize the Host–Kra–Maass characterization of the largest pronilfactor of a minimal group action.<sup>[15](https://www.impan.pl/~gutman/Nilspaces.html)</sup>

## Equidistribution and quantitative bounds

Leibman's equidistribution theorem describes limits of polynomial orbits on nilmanifolds; Green and Tao proved its quantitative form. A finite polynomial orbit \( (g(n)\Gamma) \) for \( n \in [N] \) admits a factorisation \( g = \epsilon g_0 \gamma \), where \( \epsilon(n) \) is smooth, \( (\gamma(n)\Gamma) \) is periodic and rational, and \( (g_0(n)\Gamma) \) is uniformly distributed up to error \( \delta \) inside a subnilmanifold \( G_0/\Gamma_0 \) for all sufficiently dense arithmetic progressions \( P \subseteq [N] \).<sup>[8](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p02-p.pdf)</sup> The bounds are uniform in \( N \) and polynomial in the error tolerance \( \delta \); in a companion paper the theorem was used to establish the Möbius and Nilsequences conjecture.<sup>[8](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p02-p.pdf)</sup>

**Nilflows.** For filiform nilflows whose toral projection has Diophantine frequency \( \alpha_1 \) of exponent \( \nu > k/2 \), ergodic averages satisfy
\[ \left| \int_0^T f \circ \phi_t^{X_\alpha}(x) \, dt \right| \le C_{\sigma,\epsilon}(\alpha_1) \|f\|_{W^\sigma(M)} \, T^{1 - \frac{1}{k(k-1)} + \epsilon} \]
for zero-average \( f \in W^\sigma(M) \).<sup>[16](https://ar5iv.labs.arxiv.org/html/2302.03618)</sup> The same work derives Weyl-sum bounds for higher-degree polynomials with a power saving comparable to that obtained from the Bourgain–Demeter–Guth proof of the Vinogradov Main Conjecture.<sup>[16](https://ar5iv.labs.arxiv.org/html/2302.03618)</sup> On compact three-dimensional homogeneous nilmanifolds, quantitative equidistribution of nilflows was proved by renormalization and invariant distributions, yielding a dynamical proof of quantitative equidistribution of \( P(n) \bmod 1 \) for quadratic polynomials \( P \) with generic leading coefficient.<sup>[17](https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/abs/equidistribution-of-nilflows-and-applications-to-theta-sums/197FB612315B434F8691B1A3BD16623F)</sup> Beyond polynomial driving, orbits \( b_1^{a_1(n)} \cdots b_k^{a_k(n)} \Gamma \) driven by smooth functions of polynomial growth from a Hardy field equidistribute on some subnilmanifold under growth-rate assumptions, with norm convergence of multiple ergodic averages following via the Host–Kra structure theorem and the Green–Tao quantitative result.<sup>[18](https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/abs/pointwise-convergence-in-nilmanifolds-along-smooth-functions-of-polynomial-growth/C1BC6E706AE3412DFC859F2D99B5D10A)</sup>

## By the numbers

Three parameters govern the quantitative theory. The step \( k \) enters the nilflow exponent \( T^{1 - 1/(k(k-1)) + \epsilon} \), whose power saving is \( 1/(k(k-1)) \).<sup>[16](https://ar5iv.labs.arxiv.org/html/2302.03618)</sup> The error \( \delta \) enters the Green–Tao bounds polynomially, uniformly in the orbit length \( N \).<sup>[8](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p02-p.pdf)</sup> In the inverse theorem, a recent result proves quasipolynomial bounds: if \( f : [N] \to \mathbb{C} \) is 1-bounded with [Gowers norm](https://www.edgepedia.org/gowers-norm) at least \( \delta \), then \( f \) correlates with a nilsequence on a nilmanifold \( G/\Gamma \) of degree \( s \) with dimension \( d \le \log(1/\delta)^{O_s(1)} \) and \( \epsilon^{-1}, K, M \le \exp(\log(1/\delta)^{O_s(1)}) \); a companion paper uses the result to improve the bounds on Szemerédi's theorem.<sup>[19](https://doi.org/10.48550/arxiv.2402.17994)</sup>

## How nilsystems compare with related systems

The hierarchy of structured systems runs from torus rotations upward. The Kronecker factor of any system is the largest abelian group rotation factor.<sup>[6](https://sites.math.northwestern.edu/~kra/papers/nil.pdf)</sup> [Connected](https://www.edgepedia.org/connected) nilsystems are built inductively as torus extensions, one layer at a time.<sup>[11](https://terrytao.wordpress.com/2008/03/09/254a-lecture-16-a-ratner-type-theorem-for-nilmanifolds/)</sup> Passing to inverse limits gives pro-nilsystems, the \( k \)-step versions of which are exactly the universal characteristic factors for \( k \)-term averages.<sup>[13](https://pure.uva.nl/ws/files/1693302/126537_08.pdf)</sup><sup> • </sup><sup>[5](https://www.ams.org/journals/jams/2007-20-01/S0894-0347-06-00532-7/)</sup> Nilspace theory reformulates the same structure combinatorially and provides an independent route to the inverse theorem.<sup>[15](https://www.impan.pl/~gutman/Nilspaces.html)</sup>

## Applications and recent developments

Nilsystem theory feeds directly into number theory. The quantitative equidistribution theorem underlies the Möbius and Nilsequences conjecture,<sup>[8](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p02-p.pdf)</sup> and the inverse theorem implies the Hardy–Littlewood prime tuples conjecture for finite-complexity linear systems.<sup>[7](https://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p11-p.pdf)</sup> A 2025 article in Algebra & Number Theory proves orthogonality of smooth numbers to nilsequences and, as a first application, establishes for \( y > N^{1/\sqrt{\log^9 N}} \) asymptotic results on the frequency with which an arbitrary finite-complexity system of shifted linear forms takes values with prescribed prime-factor conditions.<sup>[20](https://msp.org/ant/2025/19-10/ant-v19-n10-p01-s.pdf)</sup>

**Post-2023 results.** Several developments have reshaped the field since 2023:

- A 2024 paper shows that the universal characteristic factor of order \( < k+1 \) for Gowers–Host–Kra seminorms over \( \bigoplus_p \mathbb{Z}/p\mathbb{Z} \) is a factor of an inverse limit of finite-dimensional \( k \)-step nilpotent homogeneous spaces, generalizing Host–Kra (2005), Ziegler (2007), and Bergelson–Tao–Ziegler (2011, 2015); it is the first instance of studying Host–Kra factors of nonfinitely generated groups of unbounded torsion.<sup>[21](https://msp.org/apde/2024/17-7/apde-v17-n7-p05-p.pdf)</sup>
- A recent paper gives an explicit algebraic description of the Host–Kra \( \mathcal{Z}_k \)-factors of nonergodic nilsystems: there is a normal rational subgroup \( H \subseteq G \) such that \( (G/H_{k+1}\Gamma, \mu, T) \) is the \( \mathcal{Z}_k \)-factor for each \( k \), where \( H_{k+1} \) is the \( (k+1) \)-th term of the lower central series of \( H \), and the orbit closure of \( T^n x \) equals \( Hx \) for \( \mu \)-a.e. \( x \).<sup>[9](https://arxiv.org/html/2602.21021)</sup>
- A 2025 article defines CF-Nil(\( k \)) systems, strictly ergodic systems in which the maximal measurable and maximal topological \( k \)-step pro-nilfactors coincide, and proves pointwise convergence of two-dimensional averages \( \frac{1}{N^2} \sum_{m,n=0}^{N-1} \psi(m,n) \prod_{j=1}^d f_j(T^{m+jn}x) \) for any nilsequence \( \{\psi(m,n)\} \) and any \( f_1, \ldots, f_d \in L^\infty(\mu) \) in any ergodic system, along with \( L^2 \)-convergence of certain two-dimensional averages for non-commuting transformations without a zero-entropy condition.<sup>[22](https://www.aimsciences.org/article/doi/10.3934/dcds.2025170)</sup>
- A 2026 preprint proves that for a minimal nilrotation on a compact connected nilmanifold, the polynomial diagonal orbit closure associated with any finite family of integer-coefficient polynomials vanishing at the origin is connected, resolving a conjecture of Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson; the same paper yields polynomial multiple recurrence in every prescribed residue class in topological dynamics when the corresponding power of the transformation is minimal, and constructs a totally minimal nilsystem for which Leibman's lower central series identity fails.<sup>[14](https://arxiv.org/html/2608.22508)</sup>

The quasipolynomial inverse-theorem bounds already feed back into better bounds for Szemerédi's theorem.<sup>[19](https://doi.org/10.48550/arxiv.2402.17994)</sup>

## References

1. [Host, Kra, Maass: Complexity of nilsystems and systems lacking nilfactors](https://sites.math.northwestern.edu/~kra/papers/nilfactors.pdf)
2. [Nilpotent Structures in Ergodic Theory, AMS Surveys and Monographs 236](https://bookstore.ams.org/surv-236)
3. [On the maximal spectral type of Nilsystems (arXiv)](https://ar5iv.labs.arxiv.org/html/2307.07213)
4. [Leibman: survey of Host–Kra–Ziegler characteristic factors](https://people.math.osu.edu/leibman.1/papers/hkz.pdf)
5. [Ziegler: Universal characteristic factors and convergence of multiple ergodic averages, J. Amer. Math. Soc. 20 (2007)](https://www.ams.org/journals/jams/2007-20-01/S0894-0347-06-00532-7/)
6. [Host, Kra: Gowers norms, nilsystems and parallelepiped structures](https://sites.math.northwestern.edu/~kra/papers/nil.pdf)
7. [Green, Tao, Ziegler: An inverse theorem for the Gowers U^{s+1}[N]-norm, Annals of Math. 176 (2012)](https://annals.math.princeton.edu/wp-content/uploads/annals-v176-n2-p11-p.pdf)
8. [Green, Tao: The quantitative behaviour of polynomial orbits on nilmanifolds, Annals of Math. 175 (2012)](https://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p02-p.pdf)
9. [Structure and Spectrum of Nonergodic Nilsystems (arXiv)](https://arxiv.org/html/2602.21021)
10. [Host, Kra: Analysis of two step nilsequences, Ann. Inst. Fourier 58 (2008)](https://www.numdam.org/article/AIF_2008__58_5_1407_0.pdf)
11. [Tao: 254A, Lecture 16: A Ratner-type theorem for nilmanifolds](https://terrytao.wordpress.com/2008/03/09/254a-lecture-16-a-ratner-type-theorem-for-nilmanifolds/)
12. [Leibman: Nilsequences, null-sequences, and multiple correlation sequences](https://people.math.osu.edu/leibman.1/papers/msqx.pdf)
13. [UvA-DARE thesis chapter on nilsystems and pro-nilsystems](https://pure.uva.nl/ws/files/1693302/126537_08.pdf)
14. [Connectedness of polynomial diagonal orbit closures for minimal nilrotations and applications (arXiv, 2026)](https://arxiv.org/html/2608.22508)
15. [Gutman: What are nilspaces? (IMPAN)](https://www.impan.pl/~gutman/Nilspaces.html)
16. [Equidistribution of nilflows and bounds on Weyl sums (arXiv, 2023)](https://ar5iv.labs.arxiv.org/html/2302.03618)
17. [Equidistribution of nilflows and applications to theta sums, Ergodic Theory and Dynamical Systems](https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/abs/equidistribution-of-nilflows-and-applications-to-theta-sums/197FB612315B434F8691B1A3BD16623F)
18. [Pointwise convergence in nilmanifolds along smooth functions of polynomial growth, Ergodic Theory and Dynamical Systems](https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/abs/pointwise-convergence-in-nilmanifolds-along-smooth-functions-of-polynomial-growth/C1BC6E706AE3412DFC859F2D99B5D10A)
19. [Quasipolynomial bounds on the inverse theorem for the Gowers U^{s+1}[N]-norm](https://doi.org/10.48550/arxiv.2402.17994)
20. [Smooth numbers are orthogonal to nilsequences, Algebra & Number Theory 19 (2025)](https://msp.org/ant/2025/19-10/ant-v19-n10-p01-s.pdf)
21. [Host–Kra factors for ⨁_p Z/pZ actions and finite-dimensional nilpotent systems, APDE 17 (2024)](https://msp.org/apde/2024/17-7/apde-v17-n7-p05-p.pdf)
22. [CF-Nil systems and convergence of two-dimensional ergodic averages, DCDS (2025)](https://www.aimsciences.org/article/doi/10.3934/dcds.2025170)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › General discrete mathematics and discrete structures › Combinatorics › Extremal and additive combinatorics › Higher-order Fourier analysis and nilsystems*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license