M\"{o}bius disjointness for skew products on a circle and a nilmanifold

Jianya Liu; Ke Wang; Wen Huang

arxiv: 1907.01735 · v1 · pith:UUN4YSNRnew · submitted 2019-07-03 · 🧮 math.NT · math.DS

M\"{o}bius disjointness for skew products on a circle and a nilmanifold

Wen Huang , Jianya Liu , Ke Wang This is my paper

Pith reviewed 2026-05-25 10:29 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords Möbius disjointnessskew productsdistal dynamical systemsHeisenberg nilmanifoldSarnak conjecturenilmanifoldslinear disjointness

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The pith

A class of skew products on the circle and Heisenberg nilmanifold are distal and Möbius disjoint.

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

The paper establishes distality for a class of skew products defined on the product space consisting of the unit circle and the three-dimensional Heisenberg nilmanifold. It then demonstrates that the Möbius function is linearly disjoint from the action of these skew products. This result confirms Sarnak's Möbius Disjointness Conjecture for the systems in question. The proof relies on the specific structure of these spaces to establish the necessary dynamical properties. Readers interested in the intersection of number theory and ergodic theory would find this verification of the conjecture relevant.

Core claim

Let T be the unit circle and Γ∖G the 3-dimensional Heisenberg nilmanifold. We prove that a class of skew products on T × Γ∖G are distal, and that the Möbius function is linearly disjoint from these skew products. This verifies the Möbius Disjointness Conjecture of Sarnak.

What carries the argument

Distality of the skew products on T × Γ∖G, which is used to establish their linear disjointness from the Möbius function.

If this is right

These skew products satisfy Sarnak's Möbius Disjointness Conjecture.
The Möbius function has vanishing correlations with continuous functions on these systems.
The systems are zero-entropy dynamical systems due to distality.
The result applies specifically to skew products constructed using the geometry of the circle and the nilmanifold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach may extend the known cases of the conjecture to other product systems involving nilmanifolds.
It highlights the role of geometric constructions in proving disjointness properties.

Load-bearing premise

The skew products are restricted to a class whose distality follows directly from the geometry of the circle and the 3-dimensional Heisenberg nilmanifold.

What would settle it

A specific skew product in the class where the limit of the average of the Möbius function times a continuous function along the orbit does not equal zero would disprove the disjointness.

read the original abstract

Let $\mathbb{T}$ be the unit circle and $\Gamma \backslash G$ the $3$-dimensional Heisenberg nilmanifold. We prove that a class of skew products on $\mathbb{T} \times \Gamma \backslash G$ are distal, and that the M\"{o}bius function is linearly disjoint from these skew products. This verifies the M\"{o}bius Disjointness Conjecture of Sarnak.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves Möbius disjointness for a specific class of skew products on the circle and 3D Heisenberg nilmanifold, adding one more concrete case to Sarnak's conjecture.

read the letter

The paper proves that a class of skew products on the circle times the 3D Heisenberg nilmanifold are distal, and from that gets linear disjointness with the Möbius function. This adds one more family to the list of systems where Sarnak's conjecture holds. What stands out is the use of the geometry of these particular spaces to get the distality property. That seems to be the main technical step, and it looks like it works for the skew products they consider. The result is new as stated, and there is no sign of circular reasoning or loose assumptions in the claim. The soft spot is the narrow scope. It only covers this class of skew products, not a general class of distal systems or anything close to the full conjecture. So the impact stays within the incremental category. The math appears solid based on the description, with no obvious gaps in the logic from distality to disjointness. This is the kind of paper that specialists in ergodic theory and number theory would want to see. It is worth sending to referees because it gives a concrete new case with a proof that can be checked. I would bring it to a reading group on dynamics if the group is looking at Sarnak type results. I probably would not cite it in my own work soon, as it is quite specialized. The authors show honest engagement with the problem.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that a class of skew products on the unit circle T times the 3-dimensional Heisenberg nilmanifold Γ∖G are distal and that the Möbius function is linearly disjoint from the associated dynamical systems, thereby verifying Sarnak's Möbius Disjointness Conjecture for this class.

Significance. If the central claims hold, the result supplies a new family of distal examples on this specific product space for which the conjecture is confirmed, obtained by exploiting the geometry of the circle and the nilmanifold. The reduction from distality to linear disjointness follows standard lines in the literature.

major comments (1)

[Abstract] Abstract, paragraph 1: the precise definition of the 'class of skew products' whose distality follows from the geometry of T and the Heisenberg nilmanifold is not stated; without an explicit characterization of the admissible cocycles or maps, the scope of the theorem cannot be verified.

minor comments (1)

[Abstract] The abstract is extremely terse and omits any indication of the proof strategy or the key geometric ingredients; expanding it by one or two sentences would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive comment on the abstract. We address the point below and will incorporate the suggested clarification in the revised manuscript.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 1: the precise definition of the 'class of skew products' whose distality follows from the geometry of T and the Heisenberg nilmanifold is not stated; without an explicit characterization of the admissible cocycles or maps, the scope of the theorem cannot be verified.

Authors: We agree that the abstract would benefit from a more explicit characterization of the class. In the revised version we will expand the first paragraph of the abstract to state the precise conditions on the cocycles (specifically, the admissible continuous maps from T to the Heisenberg group satisfying the distality criterion derived from the geometry of T and Γ∖G, as defined in Section 2 of the manuscript). This will make the scope of the theorem immediately verifiable while preserving the original results. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes distality of a specific class of skew products on T × Γ∖G directly from the geometry of the circle and 3-dimensional Heisenberg nilmanifold, then derives Möbius linear disjointness as a consequence. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the derivation relies on standard dynamical systems techniques and is self-contained against external benchmarks such as Sarnak's conjecture.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard facts about distal dynamical systems, the Möbius function, and nilmanifold geometry drawn from prior literature rather than new postulates introduced in this paper.

axioms (1)

standard math Standard properties of distal dynamical systems and the Möbius function from ergodic theory and analytic number theory
Invoked to establish distality and linear disjointness for the skew products.

pith-pipeline@v0.9.0 · 5596 in / 1163 out tokens · 38749 ms · 2026-05-25T10:29:47.351333+00:00 · methodology

discussion (0)

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ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Let T be the unit circle and Γ∖G the 3-dimensional Heisenberg nilmanifold. We prove that a class of skew products on T×Γ∖G are distal...

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Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

[1]

Approximations on C(T × Γ\G) 4

work page
[2]

Theorem 1.1 for rational α 6

work page
[3]

Rational approximations of α and further analysis 10

work page
[4]

Measure complexity 14

work page
[5]

Theorem 1.1 for irrational α 16

work page
[6]

Appendix I: preliminaries on nilmanifolds and the Mal’ce v basis 23

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[7]

Appendix II: the distality of ( T × Γ\G, T ) 25 Reference 27

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The behavior of µ is central in the theory of prime numbers

Introduction Let µ(n) be the M¨ obius function, that is µ(n) is 0 if n is not square-free, and is ( − 1)k if n is a product of k distinct primes. The behavior of µ is central in the theory of prime numbers. Let ( X,T ) be a ﬂow, namely X is a compact metric space and T : X → X a continuous map. We say that µ is linearly disjoint from ( X,T ) if lim N →∞ 1...

work page
[9]

Date: July 4, 2019

states that the function µ is linearly disjoint from every ( X,T ) whose entropy is 0. Date: July 4, 2019. 2000 Mathematics Subject Classiﬁcation. 37A45, 11L03, 11N37. Key words and phrases. The M¨ obius function, distal ﬂow, skew product, nilmanifold, measure complexity. 1 This conjecture has been proved for many cases, and we refer to the survey paper [...

work page 2019
[10]

It is therefore possible to generalize our Theorem 1.1 t o the case of absolutely continuousϕ and ψ by similar arguments

has studied the M¨ obius disjointness for skew products (1.2) o n T2 whereh is absolutely continuous. It is therefore possible to generalize our Theorem 1.1 t o the case of absolutely continuousϕ and ψ by similar arguments. Notations. We list some notations that we use in the paper. We write e(x) for e2πix , and write ∥x∥ for the distance between x and th...

work page
[11]

The purpose of this sectio n is to construct a subset of C(T × Γ\G), which spans a C-linear subspace that is dense in C(T × Γ\G)

Approximations on C(T × Γ\G) LetG be the 3-dimensional Heisenberg group with the cocompact discret e subgroup Γ, and Γ\G the 3-dimensional Heisenberg nilmanifold. The purpose of this sectio n is to construct a subset of C(T × Γ\G), which spans a C-linear subspace that is dense in C(T × Γ\G). A basic reference for this section is Tolimieri [23]. For intege...

work page
[12]

In view of Proposition 2.3, we should separately consider two cases, namely f ∈ A and f ∈ B

Theorem 1.1 for rational α In this section, we prove Theorem 1.1 for rational α . In view of Proposition 2.3, we should separately consider two cases, namely f ∈ A and f ∈ B . The case f ∈ B can be reduced to the case of skew products on T2 which is already known. The other case f ∈ A will be handled by Fourier analysis and a classical result of Hua. We b...

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[13]

In this section, we will decompose ϕ (t),ϕ 2(t) and ψ (t) into the sum of resonant and non-resonant parts, and investiga te them separately

Rational approximations of α and further analysis From now on, we assume that α is irrational. In this section, we will decompose ϕ (t),ϕ 2(t) and ψ (t) into the sum of resonant and non-resonant parts, and investiga te them separately. For simplicity we write η(t) :=ϕ 2(t). Let α = [0;a1,a 2,...,a k,... ] = 1 a1 + 1 a2+ 1 a3+... be the continued fraction ...

work page
[14]

This together with (4.4) and (4.1) gives ∑ qk ≤| m|<qk+1 qk |m |a(m)| ∥mα ∥ ≪ ∑ d≥ 1 (dqk)− 2B(dqk+1) ≪ q−B k ∑ d≥ 1 d− 2B+1 ≪ q−B k , 12 where we have applied qk+1 ≤ qB k

(4.5) So ∥mα ∥ is actually equal to |d|∥qkα ∥. This together with (4.4) and (4.1) gives ∑ qk ≤| m|<qk+1 qk |m |a(m)| ∥mα ∥ ≪ ∑ d≥ 1 (dqk)− 2B(dqk+1) ≪ q−B k ∑ d≥ 1 d− 2B+1 ≪ q−B k , 12 where we have applied qk+1 ≤ qB k . Hence S2 is also absolutely convergent. The proof is complete. □ Sinceϕ is assumed to be C ∞ -smooth, we have ˆϕ (m) ≪ | m|− 2B for any ...

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[15]

In this section, we will collect some concepts and fa cts from [11] without proof

Measure complexity To prove Theorem 1.1 for irrational α , we will use the concept of measure complexity introduced in [11]. In this section, we will collect some concepts and fa cts from [11] without proof. Let (X,T ) be a ﬂow, and M(X,T ) the set of all T -invariant Borel probability measures onX. A metric d on X is called compatible if the topology ind...

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Proposition 6.1

Theorem 1.1 for irrational α The purpose of this section is to prove the next result. Proposition 6.1. Let (T × Γ\G,T ) be as in Theorem 1.1 with α irrational. Then the measure complexity of (T × Γ\G,T,ρ ) is sub-polynomial for any ρ ∈ M(T × Γ\G,T ). Before proving Proposition 6.1, we need to choose a proper metric o n T × Γ\G. The following facts can be ...

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16 It can be proved that dΓ\ G is indeed a metric on Γ \G

: g′ 1,g ′ 2 ∈ G, Γg1 = Γg′ 1, Γg2 = Γg′ 2}. 16 It can be proved that dΓ\ G is indeed a metric on Γ \G. Since dG is left-invariant, we also have dΓ\ G(Γg1, Γg2) = inf γ ∈ Γ dG(g1,γg 2). (6.3) Finally, we takedT to be the canonical Euclidean metric on T, andd =dT× Γ\ G thel∞ -product metric of dT and dΓ\ G given by d((t1, Γg1), (t2, Γg2)) = max(dT(t1,t 2),...

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Let G be a connected, simply connected Lie group

Appendix I: preliminaries on nilmanifolds and the Mal’cev ba sis Deﬁnition 7.1 (Nilmanifold). Let G be a connected, simply connected Lie group. The identity element ofG is denoted by idG. A ﬁltrationG• onG is a sequence of closed connected subgroups G =G0 =G1 ⊇ G2 ⊇ · · · ⊇Gd ⊇ Gd+1 = {idG} satisfying [Gi,G j] ⊂ Gi+j for all integers i,j ≥ 0. The degree o...

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It can be proved that dΓ\ G is indeed a metric on Γ \G

: g′ 1,g ′ 2 ∈ G, Γg1 = Γg′ 1, Γg2 = Γg′ 2}. It can be proved that dΓ\ G is indeed a metric on Γ \G. Since dG is left-invariant, we also have dΓ\ G(Γg1, Γg2) = inf γ ∈ Γ dG(g1,γg 2). 24 Deﬁnition 7.4 (Rationality of a Mal’cev basis) . Let Γ \G be a m-dimensional nilmanifold and let Q> 0. A Mal’cev basis X = {X1,X 2,...,X m} for Γ \G is called Q-rational i...

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Theorem 8.1

Appendix II: the distality of (T × Γ\G,T ) The purpose of this section is to establish the following theorem that implies the distality of the ﬂow ( T × Γ\G,T ). Theorem 8.1. Let T be the unit circle and Γ\G the 3-dimensional Heisenberg nilmanifold. Letα ∈ [0, 1) and let ϕ 1,ϕ 2,ψ beC ∞ -smooth periodic functions with period 1. Denote by S the skew produc...

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Similarly, for k suﬃciently large, r2 k − s2 k is an integral constant b satisfying b =g2 2 − g2

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So for k suﬃciently large we have (r3 k +g3 1 +h3 k +g1 1r2 k +h1 kr2 k +g2 1h1 k) − (s3 k +g3 2 +h3 k +g1 2s2 k +h1 ks2 k +g2 2h1 k) = (r3 k − s3 k) + (g3 1 − g3

Now since for large k, the x,y -components of rkg1hk and skg2hk are equal, by (8.4) and the deﬁnition of κ, the diﬀerence between their z-components tends to zero as well. So for k suﬃciently large we have (r3 k +g3 1 +h3 k +g1 1r2 k +h1 kr2 k +g2 1h1 k) − (s3 k +g3 2 +h3 k +g1 2s2 k +h1 ks2 k +g2 2h1 k) = (r3 k − s3 k) + (g3 1 − g3

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+g1 1r2 k − g1 2s2 k = (r3 k − s3 k) + (g3 1 − g3

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+g1 1r2 k − (g1 1 +a)s2 k = (r3 k − s3 k) + (g3 1 − g3

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Again, since r3 k − s3 k − as2 k ∈ Z, there exists an integral constant c such that c = r3 k − s3 k − as2 k for large k and c satisﬁes c = g3 2 − g3 1 − bg1

+g1 1b − as2 k which approaches 0 as k → ∞ . Again, since r3 k − s3 k − as2 k ∈ Z, there exists an integral constant c such that c = r3 k − s3 k − as2 k for large k and c satisﬁes c = g3 2 − g3 1 − bg1

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This is a contradiction, and the theorem is proved

As a 26 consequence, we have found a,b,c ∈ Z such that     g1 2 =g1 1 +a, g2 2 =g2 1 +b, g3 2 =g3 1 +bg1 1 +c, which implies Γg2 =   1 g2 2 g3 2 0 1 g1 2 0 0 1   = Γ   1 b c 0 1 a 0 0 1     1 g2 1 g3 1 0 1 g1 1 0 0 1   = Γg1. This is a contradiction, and the theorem is proved. □ Acknowledgements. The ﬁrst author is partially supported by ...

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[1] [1]

Approximations on C(T × Γ\G) 4

work page

[2] [2]

Theorem 1.1 for rational α 6

work page

[3] [3]

Rational approximations of α and further analysis 10

work page

[4] [4]

Measure complexity 14

work page

[5] [5]

Theorem 1.1 for irrational α 16

work page

[6] [6]

Appendix I: preliminaries on nilmanifolds and the Mal’ce v basis 23

work page

[7] [7]

Appendix II: the distality of ( T × Γ\G, T ) 25 Reference 27

work page

[8] [8]

The behavior of µ is central in the theory of prime numbers

Introduction Let µ(n) be the M¨ obius function, that is µ(n) is 0 if n is not square-free, and is ( − 1)k if n is a product of k distinct primes. The behavior of µ is central in the theory of prime numbers. Let ( X,T ) be a ﬂow, namely X is a compact metric space and T : X → X a continuous map. We say that µ is linearly disjoint from ( X,T ) if lim N →∞ 1...

work page

[9] [9]

Date: July 4, 2019

states that the function µ is linearly disjoint from every ( X,T ) whose entropy is 0. Date: July 4, 2019. 2000 Mathematics Subject Classiﬁcation. 37A45, 11L03, 11N37. Key words and phrases. The M¨ obius function, distal ﬂow, skew product, nilmanifold, measure complexity. 1 This conjecture has been proved for many cases, and we refer to the survey paper [...

work page 2019

[10] [10]

It is therefore possible to generalize our Theorem 1.1 t o the case of absolutely continuousϕ and ψ by similar arguments

has studied the M¨ obius disjointness for skew products (1.2) o n T2 whereh is absolutely continuous. It is therefore possible to generalize our Theorem 1.1 t o the case of absolutely continuousϕ and ψ by similar arguments. Notations. We list some notations that we use in the paper. We write e(x) for e2πix , and write ∥x∥ for the distance between x and th...

work page

[11] [11]

The purpose of this sectio n is to construct a subset of C(T × Γ\G), which spans a C-linear subspace that is dense in C(T × Γ\G)

Approximations on C(T × Γ\G) LetG be the 3-dimensional Heisenberg group with the cocompact discret e subgroup Γ, and Γ\G the 3-dimensional Heisenberg nilmanifold. The purpose of this sectio n is to construct a subset of C(T × Γ\G), which spans a C-linear subspace that is dense in C(T × Γ\G). A basic reference for this section is Tolimieri [23]. For intege...

work page

[12] [12]

In view of Proposition 2.3, we should separately consider two cases, namely f ∈ A and f ∈ B

Theorem 1.1 for rational α In this section, we prove Theorem 1.1 for rational α . In view of Proposition 2.3, we should separately consider two cases, namely f ∈ A and f ∈ B . The case f ∈ B can be reduced to the case of skew products on T2 which is already known. The other case f ∈ A will be handled by Fourier analysis and a classical result of Hua. We b...

work page

[13] [13]

In this section, we will decompose ϕ (t),ϕ 2(t) and ψ (t) into the sum of resonant and non-resonant parts, and investiga te them separately

Rational approximations of α and further analysis From now on, we assume that α is irrational. In this section, we will decompose ϕ (t),ϕ 2(t) and ψ (t) into the sum of resonant and non-resonant parts, and investiga te them separately. For simplicity we write η(t) :=ϕ 2(t). Let α = [0;a1,a 2,...,a k,... ] = 1 a1 + 1 a2+ 1 a3+... be the continued fraction ...

work page

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This together with (4.4) and (4.1) gives ∑ qk ≤| m|<qk+1 qk |m |a(m)| ∥mα ∥ ≪ ∑ d≥ 1 (dqk)− 2B(dqk+1) ≪ q−B k ∑ d≥ 1 d− 2B+1 ≪ q−B k , 12 where we have applied qk+1 ≤ qB k

(4.5) So ∥mα ∥ is actually equal to |d|∥qkα ∥. This together with (4.4) and (4.1) gives ∑ qk ≤| m|<qk+1 qk |m |a(m)| ∥mα ∥ ≪ ∑ d≥ 1 (dqk)− 2B(dqk+1) ≪ q−B k ∑ d≥ 1 d− 2B+1 ≪ q−B k , 12 where we have applied qk+1 ≤ qB k . Hence S2 is also absolutely convergent. The proof is complete. □ Sinceϕ is assumed to be C ∞ -smooth, we have ˆϕ (m) ≪ | m|− 2B for any ...

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In this section, we will collect some concepts and fa cts from [11] without proof

Measure complexity To prove Theorem 1.1 for irrational α , we will use the concept of measure complexity introduced in [11]. In this section, we will collect some concepts and fa cts from [11] without proof. Let (X,T ) be a ﬂow, and M(X,T ) the set of all T -invariant Borel probability measures onX. A metric d on X is called compatible if the topology ind...

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Proposition 6.1

Theorem 1.1 for irrational α The purpose of this section is to prove the next result. Proposition 6.1. Let (T × Γ\G,T ) be as in Theorem 1.1 with α irrational. Then the measure complexity of (T × Γ\G,T,ρ ) is sub-polynomial for any ρ ∈ M(T × Γ\G,T ). Before proving Proposition 6.1, we need to choose a proper metric o n T × Γ\G. The following facts can be ...

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16 It can be proved that dΓ\ G is indeed a metric on Γ \G

: g′ 1,g ′ 2 ∈ G, Γg1 = Γg′ 1, Γg2 = Γg′ 2}. 16 It can be proved that dΓ\ G is indeed a metric on Γ \G. Since dG is left-invariant, we also have dΓ\ G(Γg1, Γg2) = inf γ ∈ Γ dG(g1,γg 2). (6.3) Finally, we takedT to be the canonical Euclidean metric on T, andd =dT× Γ\ G thel∞ -product metric of dT and dΓ\ G given by d((t1, Γg1), (t2, Γg2)) = max(dT(t1,t 2),...

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Let G be a connected, simply connected Lie group

Appendix I: preliminaries on nilmanifolds and the Mal’cev ba sis Deﬁnition 7.1 (Nilmanifold). Let G be a connected, simply connected Lie group. The identity element ofG is denoted by idG. A ﬁltrationG• onG is a sequence of closed connected subgroups G =G0 =G1 ⊇ G2 ⊇ · · · ⊇Gd ⊇ Gd+1 = {idG} satisfying [Gi,G j] ⊂ Gi+j for all integers i,j ≥ 0. The degree o...

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It can be proved that dΓ\ G is indeed a metric on Γ \G

: g′ 1,g ′ 2 ∈ G, Γg1 = Γg′ 1, Γg2 = Γg′ 2}. It can be proved that dΓ\ G is indeed a metric on Γ \G. Since dG is left-invariant, we also have dΓ\ G(Γg1, Γg2) = inf γ ∈ Γ dG(g1,γg 2). 24 Deﬁnition 7.4 (Rationality of a Mal’cev basis) . Let Γ \G be a m-dimensional nilmanifold and let Q> 0. A Mal’cev basis X = {X1,X 2,...,X m} for Γ \G is called Q-rational i...

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Theorem 8.1

Appendix II: the distality of (T × Γ\G,T ) The purpose of this section is to establish the following theorem that implies the distality of the ﬂow ( T × Γ\G,T ). Theorem 8.1. Let T be the unit circle and Γ\G the 3-dimensional Heisenberg nilmanifold. Letα ∈ [0, 1) and let ϕ 1,ϕ 2,ψ beC ∞ -smooth periodic functions with period 1. Denote by S the skew produc...

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Similarly, for k suﬃciently large, r2 k − s2 k is an integral constant b satisfying b =g2 2 − g2

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So for k suﬃciently large we have (r3 k +g3 1 +h3 k +g1 1r2 k +h1 kr2 k +g2 1h1 k) − (s3 k +g3 2 +h3 k +g1 2s2 k +h1 ks2 k +g2 2h1 k) = (r3 k − s3 k) + (g3 1 − g3

Now since for large k, the x,y -components of rkg1hk and skg2hk are equal, by (8.4) and the deﬁnition of κ, the diﬀerence between their z-components tends to zero as well. So for k suﬃciently large we have (r3 k +g3 1 +h3 k +g1 1r2 k +h1 kr2 k +g2 1h1 k) − (s3 k +g3 2 +h3 k +g1 2s2 k +h1 ks2 k +g2 2h1 k) = (r3 k − s3 k) + (g3 1 − g3

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+g1 1r2 k − g1 2s2 k = (r3 k − s3 k) + (g3 1 − g3

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+g1 1r2 k − (g1 1 +a)s2 k = (r3 k − s3 k) + (g3 1 − g3

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Again, since r3 k − s3 k − as2 k ∈ Z, there exists an integral constant c such that c = r3 k − s3 k − as2 k for large k and c satisﬁes c = g3 2 − g3 1 − bg1

+g1 1b − as2 k which approaches 0 as k → ∞ . Again, since r3 k − s3 k − as2 k ∈ Z, there exists an integral constant c such that c = r3 k − s3 k − as2 k for large k and c satisﬁes c = g3 2 − g3 1 − bg1

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This is a contradiction, and the theorem is proved

As a 26 consequence, we have found a,b,c ∈ Z such that     g1 2 =g1 1 +a, g2 2 =g2 1 +b, g3 2 =g3 1 +bg1 1 +c, which implies Γg2 =   1 g2 2 g3 2 0 1 g1 2 0 0 1   = Γ   1 b c 0 1 a 0 0 1     1 g2 1 g3 1 0 1 g1 1 0 0 1   = Γg1. This is a contradiction, and the theorem is proved. □ Acknowledgements. The ﬁrst author is partially supported by ...

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