1 Introduction
Let $(X, T)$ be a topological dynamic system, where X is a compact metric space and $T: X\to X$ a continuous map. The Möbius function $\mu : \mathbb {N} \rightarrow \{-1, 0, 1\}$ is defined by $\mu (1)=1$ , $\mu (n) =(-1)^k$ when n is the product of k distinct primes and $\mu (n) = 0$ otherwise. The Sarnak conjecture [Reference Sarnak12, Reference Sarnak13] (also called the Möbius disjointness conjecture) is the following statement.
Conjecture 1.1. Let $(X, T)$ be a topological dynamical system with zero topological entropy. Then
In recent years, there have been many results supporting the Möbius disjointness conjecture (see the comprehensive survey [Reference Ferenczi, Kułaga-Przymus, Lemańczyk, Ferenczi, Kułaga-Przymus and Lemańczyk2]). Here we discuss only the historical developments that are more relevant to this paper.
We consider the skew product $(\mathbb {T}^2, T)$ , where $\mathbb {T}^2=(\mathbb {R}/\mathbb {Z})^2$ , T is the transformation $T: (x,y) \mapsto (x+\alpha ,y+h(x))$ and $h \colon \mathbb {T} \rightarrow \mathbb {T}$ is a continuous function. Since T is distal and distal systems have zero topological entropy [Reference Parry, Auslander and Gottschalk11], T should satisfy the Möbius disjointness conjecture. In fact, skew products are building blocks of distal flows according to Furstenberg’s structure theorem of minimal distal flows [Reference Furstenberg3].
The skew product was first considered by Liu and Sarnak [Reference Liu and Sarnak10]. They proved Conjecture 1.1 for h analytic and $\lvert \widehat {h}(m)\rvert \gg e^{-\tau \lvert m\rvert }$ for some $\tau>0$ , where $\widehat {h}(m)$ is the mth Fourier coefficient of h. After that, Wang [Reference Wang16] removed the additional condition and hence obtained the Möbius disjointness conjecture for all analytic skew products on $(\mathbb {T}^2,T)$ . Huang et al. [Reference Huang, Wang and Ye7] improved the result, assuming that h is $C^\infty $ -smooth. Recently, Kanigowski et al. [Reference Kanigowski, Lemańczyk and Radziwiłł8] proved it when h is $C^{2+\varepsilon}$ -smoothand $\widehat {h}(0)=0$ , and de Faveri [Reference de Faveri1] proved it just assuming that h is $C^{1+\varepsilon }$ -smooth. Kułaga-Przymus and Lemańczyk [Reference Kułaga-Przymus and Lemańczyk9] proved that, if h is $C^{1+\varepsilon }$ -smooth and $\alpha $ is topological generic, then the Möbius disjointness conjecture is true. In 2020, Wang and Yao [Reference Wang and Yao15] proved strong orthogonality between the Möbius function and the skew products when h is $C^{3+\varepsilon }$ -smooth and $\alpha $ is measure-theoretically generic. A nilsystem is also a distal flow and the Möbius disjointness conjecture for nilsystems was proved by Green and Tao [Reference Green and Tao5].
Recently, Huang et al. [Reference Huang, Liu and Wang6] considered the Möbius disjointness conjecture for skew products on $\mathbb {T} \times {\Gamma \backslash G}$ , where G is a $3$ -dimensional Heisenberg group with the cocompact discrete subgroup $\Gamma $ , namely
Then ${\Gamma \backslash G}$ is the $3$ -dimensional Heisenberg nilmanifold. Their main result is as follows.
Theorem 1.2 (Huang et al., [Reference Huang, Liu and Wang6]).
Let $\mathbb {T}$ be the unit circle and ${\Gamma \backslash G}$ the $3$ -dimensional Heisenberg nilmanifold. Let $\alpha \in [0,1)$ and let $\varphi , \psi $ be $C^{\infty }$ -smooth periodic functions from $\mathbb {R}$ to $\mathbb {R}$ with period $1$ such that
Let the skew product T on ${\mathbb {T} \times \Gamma \backslash G}$ be given by
Then, for any $(t_0,\Gamma g_0) \in {\mathbb {T} \times \Gamma \backslash G}$ and any $f \in C({\mathbb {T} \times \Gamma \backslash G})$ ,
Remark 1.3. In [Reference Huang, Liu and Wang6], it is also proved that the skew product (1.1) on $\mathbb {T}\times {\Gamma \backslash G}$ is a distal flow.
As mentioned by Huang et al. [Reference Huang, Liu and Wang6], we can combine the ideas from [Reference de Faveri1, Reference Huang, Liu and Wang6, Reference Kanigowski, Lemańczyk and Radziwiłł8] to give the following new theorem, which improves the result in Theorem 1.2.
Theorem 1.4. Let $\mathbb {T}$ be the unit circle and ${\Gamma \backslash G}$ the $3$ -dimensional Heisenberg nilmanifold. Let $\alpha \in [0,1)$ . Let $\phi :\mathbb {T}\rightarrow \mathbb {R}$ be a $C^{2+\varepsilon }$ -smooth function and let $\psi : \mathbb {T} \rightarrow \mathbb {R}$ be a $C^{1+\varepsilon }$ -smooth function. Moreover, assume that $\hat {\psi }(0) = \hat {\phi }(0)=0$ . Let the skew product T on ${\mathbb {T} \times \Gamma \backslash G}$ be given by
Then, for any $(t_0,\Gamma g_0) \in {\mathbb {T} \times \Gamma \backslash G}$ and any $f \in C({\mathbb {T} \times \Gamma \backslash G})$ ,
Remark 1.5. From the definition of T in (1.2), we can easily compute
where $\Phi (n,t)=\sum _{l=0}^{n-1}\phi (l\alpha +t)$ , $\Psi (n,t)=\sum _{l=0}^{n-1}\psi (l\alpha +t)$ and $H(n,t)=\sum _{l=0}^{n-1}\phi ^2(l\alpha +t)$ .
Notation. For a topological dynamical system, let $M(X,T)$ be the set of T-invariant Borel probability measures on X. We write $e(x)$ for $e^{2\pi i x}$ and $\lVert x\rVert = \min _{n \in \mathbb {Z}} \lvert x-n\rvert $ for the distance between x and the nearest integer. For positive A, the notation $B=O(A)$ or $B \ll A$ means that there exists a positive constant c such that $\lvert B\rvert \leq cA$ . If the constant c depends on a parameter b, we write $B=O_{b}(A)$ or $B \ll _{b} A$ . The notation $A \asymp B$ means that $A \ll B$ and $B \ll A$ . For a topological space X, we use $C(X)$ to denote the set of all continuous complex-valued functions on X.
2 Theorem 1.4 for rational $\alpha $
Let G be the $3$ -dimensional Heisenberg group, with the cocompact discrete subgroup $\Gamma $ , and ${\Gamma \backslash G}$ the $3$ -dimensional Heisenberg nilmanifold. We follow the papers of Huang et al. [Reference Huang, Liu and Wang6, Section 2] and Tolimieri [Reference Tolimieri14] for the following lemmas.
For $0 \leq j \leq m-1$ , $j,m\in \mathbb {Z}$ , we define the functions $\phi _{mj}$ and $\phi _{mj}^{*}$ on G by
and
We can check that $\phi _{mj}$ and $\phi _{mj} ^*$ are $\Gamma $ -invariant, that is,
for any $g \in G$ and for any $\gamma \in \Gamma .$ Thus, $\phi _{mj}$ and $\phi _{mj}^*$ can be regarded as functions on the nilmanifold ${\Gamma \backslash G}$ .
Lemma 2.1 [Reference Huang, Liu and Wang6, Proposition 2.3].
Let $\mathcal {A}$ be the subset of $f \in C({\mathbb {T} \times \Gamma \backslash G})$ such that
where $\xi _1,\xi _2,\xi _3 \in \mathbb {Z}$ and $\phi = \phi _{mj}, \overline {\phi }_{mj}, \phi _{mj}^{*}$ or $\overline {\phi }^{*}_{mj}$ for some $0 \leq j \leq m-1$ . Here, $\overline {\phi }_{mj}$ and $\overline {\phi }^{*}_{mj}$ stand for the complex conjugates of $\phi _{mj}$ and $\phi _{mj}^{*}$ , respectively. Let $\mathcal {B}$ be the subset of $f \in C({\mathbb {T} \times \Gamma \backslash G})$ satisfying $f: (t,\Gamma g) \mapsto f_1(t)f_2(\Gamma g)$ with $ f_1\in C(\mathbb {T})$ and $f_2 \in C_0({\Gamma \backslash G})$ . Then the $\mathbb {C}$ -linear subspace spanned by $\mathcal {A} \cup \mathcal {B}$ is dense in $C({\mathbb {T} \times \Gamma \backslash G})$ .
To prove Theorem 1.4 for rational $\alpha $ , we need to consider two cases: $f\in \mathcal {A}$ and $f\in \mathcal {B}$ . We use the argument given by Huang et al. [Reference Huang, Liu and Wang6, Section 3].
Lemma 2.2 [Reference de Faveri1, Theorem 1].
Let $\alpha \in \mathbb {R}$ and let $h \colon \mathbb {T} \rightarrow \mathbb {T}$ be $C^{1+\varepsilon }$ -smooth. Define the skew product $T \colon \mathbb {T}^2 \rightarrow \mathbb {T}^2$ by
Then the Möbius disjointness conjecture holds for this $(\mathbb {T}^2,T)$ .
Corollary 2.3. Let $\alpha \in \mathbb {R}$ and let $h_1,h_2 \colon \mathbb {T} \rightarrow \mathbb {T}$ be $C^{1+\varepsilon }$ -smooth functions. Let $T \colon \mathbb {T}^3 \rightarrow \mathbb {T}^3$ be given by
Then the Möbius disjointness conjecture holds for $(\mathbb {T}^3,T)$ .
Proof. The proof is similar to the proof of [Reference Huang, Liu and Wang6, Corollary 3.2]; at the last step, we use Lemma 2.2.
Lemma 2.4. Let $\mathcal {B} \subset C({\mathbb {T} \times \Gamma \backslash G})$ be as in Lemma 2.1. Let T be as in Theorem 1.4 and let $f \in \mathcal {B}$ . Then, for any $(t_0,\Gamma g_0) \in {\mathbb {T} \times \Gamma \backslash G}$ ,
Proof. The proof is similar to the proof of [Reference Huang, Liu and Wang6, Corollary 3.2] but using Corollary 2.3.
Lemma 2.5 [Reference Huang, Liu and Wang6, Proposition 3.5].
Let $({\mathbb {T} \times \Gamma \backslash G},T)$ be as in Theorem 1.4 and assume ${\alpha \in \mathbb {Q} \cap [0,1)}$ . Let $\mathcal {A}$ be as in Proposition 2.1. Then, for any $(t_0,\Gamma g_0) \in {\mathbb {T} \times \Gamma \backslash G}$ , any $f\in \mathcal {A}$ and any $A>0$ ,
where the implied constant depends on A and $\alpha $ only.
Proposition 2.6. Theorem 1.4 holds for rational $\alpha $ .
3 Theorem 1.4 for irrational $\alpha $
Before proving Theorem 1.4 for irrational $\alpha $ , we choose a proper metric on ${\mathbb {T} \times \Gamma \backslash G}$ . This can be found in [Reference Green and Tao4, Sections 2 and 5] and [Reference Huang, Liu and Wang6, Section 5]. Let the closed connected subgroups $G= G_1 \supseteq G_2 \supseteq G_3= \{\mathrm {id}_G\}$ be the lower central series filtration $G_{\bullet }$ on G. Let
Then $\mathcal {X}=\{X_1,X_2,X_3\}$ is a Mal’cev basis adapted to $G_{\bullet }$ . The corresponding Mal’cev coordinate map $\kappa : G \rightarrow \mathbb {R}^3$ is given by
The metric $d_G:G\times G\rightarrow \mathbb {R}_{\geq 0}$ on G is defined to be the largest metric such that $d_G(g_1,g_2) \leq \lvert \kappa (g_1 ^{-1} g_2)\rvert $ , where $\lvert \cdot \rvert $ is the $l^{\infty }$ -norm on $\mathbb {R}^3$ . This metric can be more explicitly expressed as
from which we can see that $d_G$ is left-invariant. By (3.1),
provided that $x,y \in [0,1)$ . The above metric on G descends to a metric on ${\Gamma \backslash G}$ given by
It can be proved that $d_{{\Gamma \backslash G}}$ is indeed a metric on ${\Gamma \backslash G}$ . Since $d_{G}$ is left-invariant, we also have
Finally, we take $d_{\mathbb {T}}$ to be the canonical Euclidean metric on $\mathbb {T}$ , and $d=d_{{\mathbb {T} \times \Gamma \backslash G}}$ the $l^{2}$ -product metric of $d_{\mathbb {T}}$ and $d_{{\Gamma \backslash G}}$ given by
To prove Theorem 1.4 for irrational $\alpha $ , we use the recent new ideas of Kanigowski et al. [Reference Kanigowski, Lemańczyk and Radziwiłł8, Theorem 1] and de Faveri [Reference de Faveri1] to give the following lemmas.
Lemma 3.1. Let $0<\varepsilon < {1}/{100}$ and $\alpha $ be an irrational number. Let $\phi :\mathbb {T}\rightarrow \mathbb {R}$ be a $C^{2+\varepsilon }$ -smooth function and let $\psi : \mathbb {T} \rightarrow \mathbb {R}$ be a $C^{1+\varepsilon }$ -smooth function. Assume that $\hat {\psi }(0) = \hat {\phi }(0)=0$ . Then there exists an unbounded sequence $\{r_n\}_{n\geq 1}$ of $\mathbb {N}$ such that
for any $\nu \in M(\mathbb {T}\times {\Gamma \backslash G}, T)$ .
Lemma 3.2 [Reference Kanigowski, Lemańczyk and Radziwiłł8].
Let $(X,T)$ be a topological dynamical system and assume that for every $\nu \in M(X,T)$ , $(X,\mathcal {B},\nu ,T)$ satisfies the PR rigidity condition: there exists a linearly dense set $\mathcal {F} \subset C(X)$ such that for each $f \in \mathcal {F}$ , we can find $\delta> 0$ and a sequence $\{q_n\}_{n\geq 1}$ satisfying
Then, $(X,T)$ is Möbius disjoint, that is,
Proof of Theorem 1.4 assuming Lemma 3.1.
The case that $\alpha $ is rational has been proved in Proposition 2.6. So we assume that $\alpha $ is irrational. Let $\{r_n\}_{n\geq 1}$ be the sequence from Lemma 3.1. For any $\nu \in M(\mathbb {T}\times {\Gamma \backslash G}, T)$ ,
Here we use the triangle inequality and the T-invariance of $\nu $ . If f is also Lipschitz continuous, then, by Lemma 3.1,
for every $\nu \in M(\mathbb {T}\times {\Gamma \backslash G}, T)$ , which satisfies the condition of Lemma 3.2 and Theorem 1.4 is proved.
4 Proof of Lemma 3.1
In this section, we prove Lemma 3.1 and hence finish the proof of our main result. We assume that $\alpha $ is irrational. Let
be the continued fraction expansion of $\alpha $ . Let $p_k/q_k=[0;a_1,a_2, \ldots ,a_k]$ be the kth convergent of $\alpha $ . Then we have the following well-known properties of $p_k/q_k$ .
Lemma 4.1. Let $\alpha \in [0,1)$ be an irrational number and $p_k/q_k$ the kth convergent of $\alpha $ . Then:
-
(i) $p_0=0, p_1=1$ and $p_{k+2}=a_{k+2} p_{k+1}+ p_{k}$ for any $k \geq 0$ ; $q_0=1, q_1=a_1$ and $q_{k+2}=a_{k+2} q_{k+1}+ q_{k}$ for all $k \geq 0$ ;
-
(ii) ${1}/{(q_k+q_{k+1})} < \lVert q_k \alpha \rVert < {1}/{q_{k+1}}$ for any $k\geq 1$ ;
-
(iii) if $0<q<q_{k+1}$ , then $\lVert q_k\alpha \rVert \leq \lVert q\alpha \rVert $ .
We also need the following two lemmas.
Lemma 4.2. For $\alpha \in \mathbb {R}\setminus \mathbb {Q}$ and $k \geq 2$ ,
Proof. The first part is from [Reference de Faveri1, Lemma 2] and we can similarly get the second.
Lemma 4.3. For $\alpha \in \mathbb {R}\setminus \mathbb {Q}$ , $k \geq 1$ and $1 \leq c \leq q_k$ ,
Proof. The first part is from [Reference de Faveri1, Lemma 3] and we can similarly get the second.
Now we are ready to give the proof of Lemma 3.1.
Proof of Lemma 3.1.
Let $\Phi (n,t)=\sum _{l=0}^{n-1}\phi (l\alpha +t)$ , $\Psi (n,t)=\sum _{l=0}^{n-1}\psi (l\alpha +t)$ and $H(n,t)=\sum _{l=0}^{n-1}\phi ^2(l\alpha +t)$ , then
We consider
Therefore,
So we should choose a sequence $\{r_n\}_{n\geq 1}$ of $\mathbb {N}$ such that $\lVert r_n\alpha \rVert ^2$ , $\int _{\mathbb {T} \times {\Gamma \backslash G}}\lvert \Phi (r_n,t)\rvert ^2\,d\nu (t,\Gamma g)$ , $\int _{\mathbb {T} \times {\Gamma \backslash G}}\lvert \Psi (r_n,t)\rvert ^2\,d\nu (t,\Gamma g)$ , $\int _{\mathbb {T} \times {\Gamma \backslash G}}\lvert \Phi (r_n,t)\rvert ^4\,d\nu (t,\Gamma g)$ and $\int _{\mathbb {T} \times {\Gamma \backslash G}}\lvert H(r_n,t)\rvert ^2\,d\nu (t,\Gamma g)$ are bounded. The following argument is similar to [Reference de Faveri1, Reference Kanigowski, Lemańczyk and Radziwiłł8].
First, we consider $\int _{\mathbb {T} \times {\Gamma \backslash G}}\lvert \Phi (n,t)\rvert ^2\,d\nu (t,\Gamma g)$ . This integral is only dependent on the first coordinate t. With the projection map $\pi (t,\Gamma g)=t$ , we can rewrite the integral as
where the Borel probability measure $\pi _*\nu $ is the Lebesgue measure on $\mathbb {T}$ since $\alpha $ is irrational. Since $\phi (t)=\sum _{m\neq 0}\widehat {\phi }(m)e(mt)$ , we have
By substituting this into (4.1), we get
Now we can choose the desired sequence $\{r_n\}_{n\geq 1}$ of $\mathbb {N}$ .Temporarily choose $r_n=q_n$ , where $q_n$ is defined in Lemma 4.1. Then we break the sum in (4.2) into two sums according to $0<\lvert m\rvert <q_n$ and $\lvert m\rvert \geq q_n$ .
For the second sum, we use the fact that $\widehat {\phi }(m)\ll m^{-1-\varepsilon }(m\neq 0)$ . (We only need $\phi $ to be $C^{1+\varepsilon }$ -smooth here, but the stronger assumption that $\phi $ is $C^{2+\varepsilon }$ -smooth is needed later.) We also have $\lvert 1-e(m\alpha )\rvert \asymp \lVert m\alpha \rVert $ , $\lvert 1-e(mn\alpha )\rvert \leq 2$ and the trivial bound $\lvert {(1-e(mn\alpha ))}/{(1-e(m\alpha ))}\rvert \leq n$ . Then
For the first sum, we use the fact that $\lvert 1-e(m\alpha )\rvert \asymp \lVert m\alpha \rVert $ and $\lvert 1-e(mr_n\alpha )\rvert \asymp \lVert mq_n\alpha \rVert \leq m^2 \lVert q_n\alpha \rVert ^2<m^2q_{n+1}^{-2}$ . Then
For this sum, we consider the following two cases:
Case A. There is a subsequence $\{q_{b_n}\}_{n \geq 1}$ of $\{q_n\}_{n \geq 1}$ such that $q_{b_n + 1} \geq q_{b_n}^2$ for all $n \geq 1$ .
In this case, we replace $\{r_n\}_{n \geq 1}$ by $\{q_{b_n}\}_{n\geq 1}$ and notice that the estimate (4.3) still holds for this new $\{r_n\}_{n \geq 1}$ . By Lemma 4.2,
Case B. For all sufficiently large n, we have $q_{n+1} < q_n^2$ .
In this case, for any $0<k<n$ , Lemma 4.2 gives
We choose k with $0<k<n$ such that $q_k\in [q_n^{1/4}, q_n^{1/2}]$ , which is possible since for all sufficiently large n, we have $q_{n+1} < q_n^2$ . Then (4.4) and (4.5) give
This proves that $\int _{\mathbb {T} \times {\Gamma \backslash G}}\lvert \Phi (r_n,t)\rvert ^2\,d\nu (t,\Gamma g)\ll r_n^{-\lambda }$ for some $\lambda> \varepsilon /100$ . By the same argument, we see that with this choice of $\{r_n\}_{n \geq 1}$ , the same estimate also holds for the integrals of $\lvert \Psi (r_n,t)\rvert ^2$ .
For the first term $\lVert n\alpha \rVert $ , we have
where $b_n$ is the sequence of indices with $q_{b_n + 1} \geq q_{b_n}^2$ in Case A and $b_n = n$ in Case B.
The estimates for integrals of $\lvert H(r_n,t)\rvert ^2$ and $\lvert \Phi (r_n,t)\rvert ^4$ are similar, so we only state the differences in the proof.
For the integral of $\lvert H(r_n,t)\rvert ^2$ , we need to slightly modify the choice of $r_n$ since now the zeroth Fourier coefficient of $\eta (t) := \varphi ^2(t)$ does not vanish in general. Therefore, we should consider the extra term $\Vert r_n \hat {\eta }(0)\Vert $ . To overcome this difficulty, we follow the idea of de Faveri [Reference de Faveri1]. By Dirichlet’s approximation theorem, for any $n \geq 1$ , we can find some $s_n \leq q_n ^{\delta }$ such that $\Vert s_n q_n \hat {\eta }(0)\Vert < q_n ^{-\delta }$ where $0<\delta <\varepsilon /10$ . Then we can slightly modify the definition of $r_n$ by setting $r_n = s_n q_n$ . Since $s_n$ is quite small compared with $q_n$ , the above argument for this new choice of $r_n$ is still valid.
For the integral of $\lvert \Phi (r_n,t)\rvert ^4$ , we cannot use the Parseval identity, so we estimate it pointwise. At this point, we need the stronger smoothness for $\phi $ to obtain the desired upper bound for the Fourier coefficients. We consider the same two cases.
Case A. In this case, we have $q_{n+1} < q_n ^2$ for all sufficiently large n and $r_n = s_n q_n$ . Since $\phi $ is $C^{2+\varepsilon }$ -smooth, we have $\hat {\phi }(m) \ll \lvert m\rvert ^{-2-\varepsilon }$ for $m \neq 0$ . Thus,
For $\lvert m\rvert \geq q_n$ , by Lemma 4.3,
For $\lvert m\rvert < q_n$ ,
Now since $q_{n+1} < q_n ^2$ , there exists some $q_k \in [q_n ^{1/4}, q_n ^{1/2} ]$ . So
provided that $\lambda $ is sufficiently small.
Case B. In this case, there is a subsequene $\{q_{b_n}\}$ of $\{q_n\}$ such that $q_{b_n+1} \geq q_{b_n} ^2$ and we choose $r_n = s_{b_n}q_{b_n}$ . Then, by the same argument as in Case A,
if $\lambda $ is sufficiently small.
Finally, for $0<\lvert m\rvert < q_{b_n}$ ,
provided that $\lambda $ is sufficiently small. This proves Lemma 3.1.
Acknowledgement
We are grateful to the anonymous referees for the helpful corrections and suggestions.