Monatsh Math
DOI 10.1007/s00605-015-0808-6
The Möbius function and continuous extensions
of rotations
J. Kułaga-Przymus1,2 · M. Lemańczyk2
Received: 5 August 2014 / Accepted: 12 August 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract Let f : T → R be of class C 1+δ for some δ > 0 and let c ∈ Z. We show
that for a generic α ∈ R, the extension Tc, f : T2 → T2 of the irrational rotation
T x = x + α, given by Tc, f (x, u) = (x + α, u + cx + f (x)) (mod 1) satisfies Sarnak’s
conjecture.
Keywords The Möbius function · Sarnak’s conjecture · Skew products · Compact
group extensions · Distal flows · Joinings
Mathematics Subject Classification
37A45 · 11N37
Contents
1 Introduction . . . . . . . . . . .
2 Results . . . . . . . . . . . . . .
2.1 On the strategy of the proofs
2.2 General remarks . . . . . . .
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Communicated by H. Bruin.
Research supported by Narodowe Centrum Nauki Grant DEC-2011/03/B/ST1/00407.
B
J. Kułaga-Przymus
[email protected]
M. Lemańczyk
[email protected]
1
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland
2
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18,
87-100 Toruń, Poland
123
J. Kułaga-Przymus, M. Lemańczyk
2.3 Continuous extensions of rational rotations . . . .
2.4 Affine case . . . . . . . . . . . . . . . . . . . . .
2.4.1 Case c = 1, α ∈
/Q . . . . . . . . . . . . .
2.4.2 Case c = 0, α ∈
/Q . . . . . . . . . . . . .
2.5 Generic case: compact group extensions . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . .
Notation and basic facts . . . . . . . . . . . . . . . .
Compact group extensions: ergodicity and minimality
A remark on the KBSZ criterion for T x = x + α . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction
Recall that the Möbius function µ : N → {−1, 0, 1} is a multiplicative function1
defined as µ( p1 · . . . · pk ) = (−1)k for distinct prime numbers p j , µ(1) = 1 and
0 otherwise. Its importance is reflected
in the fact that the prime number theorem2
!
is equivalent to the condition k≤x µ(k) = o(x) and the Riemann hypothesis is
!
1
equivalent to the condition k≤x µ(k) = Oε (x 2 +ε ) for any ε > 0 (when x →
∞). The Möbius function appears to behave rather randomly and this statement was
formalized in the following conjecture of Sarnak:
Conjecture 1 ([26]) Let X be a compact metric space and let T : X → X be a
homeomorphism of zero topological entropy. Let x ∈ X , g ∈ C(X ). Then
"
n≤N
g(T n x)µ(n) = o(N ).
(1)
Whenever condition (1) is true for some T for each x ∈ X and each g ∈ C(X ), we say
that Sarnak’s conjecture holds for T or that T is disjoint with µ. We will then write
T ⊥ µ.3
Sarnak’s conjecture is known to hold in several situations, including rotations [9],
nilsystems [15], horocycle flows [7], large class of rank one maps [2,6] and certain
subclasses of dynamical systems generated by generalized Morse sequences [19],
including the classical Thue-Morse system: [1,5,8,14,16,24] and the dynamical system generated by the Rudin-Shapiro sequence [25].
One of the most fruitful tools used for proving disjointness with the Möbius function
turns out to be the following orthogonality criterion of Katai-Bourgain-Sarnak-Ziegler
(we will refer to it as KBSZ criterion).
Theorem 1.0.1 ([7,17]) Let F : N → C be a bounded sequence. Suppose that
"
n≤N
F(r n)F(sn) = o(N )
(2)
1 Function ν : N → C is said to be multiplicative if ν(mn) = ν(m)ν(n) for n, m relatively prime.
2 Recall that the prime number theorem states that π(x) = x + o( x ), where π(x) is the number of
ln x
ln x
primes less than x.
3 Notice that it suffices to show that (1) holds for a linearly dense set of continuous functions to obtain
disjointness with the Möbius function.
123
The Möbius function and continuous extensions of rotations
for any pair of sufficiently large primes r ̸= s. Then
"
n≤N
F(n)ν(n) = o(N ),
(3)
for any multiplicative function ν with |ν| ≤ 1.
In order to use this theorem for proving Conjecture 1 for a given homeomorphism
T : X → X , one takes
F(n) := g(T n x)
for n ∈ Z, x ∈ X and g ∈ C(X ).
(4)
Notice that (2) for F(n) given by (4) takes the form
"
n≤N
#
$
g T r n x g(T sn x) = o(N ),
(5)
where
⎛
⎞
%
"
"
#
$
1
1
g ⊗ g (T r × T s )n (x, x) =
g⊗gd⎝
δ(T r ×T s )n (x,x) ⎠.
2
N
N
X
n≤N
n≤N
!
Thus, we are interested in the limit measures ρ = limk→∞ N1k n≤Nk δ(T r ×T s )n (x,x)
which are clearly T r × T s -invariant. Therefore, to prove that T ⊥ µ, it suffices to
show that the following holds for T , for r, s relatively prime:
(a) The ergodic components of T r × T s are pairwise disjoint closed sets filling up
the whole space.4
(b) The ergodic components of T r × T s are uniquely ergodic (this implies that all
points are generic for T r × T s for relevant invariant measures).
(c) There exists a linearly dense set of continuous
functions F ⊂ C(X ) such that for
*
each g ∈ F, g ◦ T ̸= g, we have X 2 g ⊗ g dρ = 0 for any T r × T s -ergodic
measure ρ, whenever r, s are sufficiently large.5
We will use the strategy (a), (b), (c) to study disjointness with µ in the following
setting. Denote by T = R/Z = [0, 1) the additive circle and consider
T2 ∋ (x, y) -→ Tc, f (x, y) := (x + α, y + cx + f (x)) ∈ T2 ,
(6)
4 The proof of the main result of the paper (Theorem 1.0.3 below) says also that the ergodic decomposition
for T r × T s is the same as the decomposition into minimal components. This seems to be of independent
interest. In case of continuous compact group extensions the existence of the decomposition into minimal
components is guaranteed by a result of Auslander [4] and Ellis [10] on distal systems.
5 Notice that we do not aim to prove (5) for each g ∈ C(X ) (and each x ∈ X )—as a matter of fact (5)
is not satisfied for some continuous functions already for an irrational rotation; we provide examples in
“Appendix”, see Proposition 2.5.17. Our aim is to prove (5) for a linearly dense set of g ∈ C(X ) (and each
x ∈ X ), as it implies that (1) holds for each g ∈ C(X ) (and each x ∈ X ).
123
J. Kułaga-Przymus, M. Lemańczyk
where c ∈ Z and cx + f (x) is a lift of a continuous circle map, i.e. f : R → R is
continuous, periodic of period 1, and c is the degree of f . In other words, we consider
continuous case of the classical Anzai skew product extensions of a rotation on the
circle [3].
Liu and Sarnak in their recent paper [23] proved the following.
Theorem 1.0.2 ([23]) Let c ∈ Z and assume that f : R → R is an analytic periodic
function of period 1. Assume additionally that | +
f (m)| ≫ e−τ |m| for some τ > 0. Then
Tc, f ⊥ µ.
The technical condition | +
f (m)| ≫ e−τ |m| on the Fourier transform seems to be necessary for the methods of [23] to work. On the other hand, there is no condition on
α. Moreover, for some αs, the result is obtained using Theorem 1.0.1. Under some
additional assumptions, also a quantitative version [i.e. concerning the speed of convergence to zero in (1)] of Theorem 1.0.2 is proved in [23]. This is achieved by treating
the problem in a more direct way than applying Theorem 1.0.1.
A natural question arises whether the assumptions on f in Theorem 1.0.2 can be
relaxed. Our main result (Theorem 1.0.3 below) provides a positive answer for each
sufficiently smooth f , at the cost of reducing “for each α” in Theorem 1.0.2 to “for a
generic α”. Hence, it can be viewed as complementary to Liu-Sarnak’s result.
Theorem 1.0.3 Let c ∈ Z and assume that f : R → R is of class C 1+δ for some
δ > 0, periodic of period 1. Then, for a generic set of α, we have Tc, f ⊥ µ.
Remark 1.0.4 The question of whether an analogous result to Theorem 1.0.3 is true
for f which is only assumed to be continuous, remains open.6
Before we give the proof of Theorem 1.0.3, we first show that Conjecture 1 holds
in the following two natural cases.
1. For an arbitrary continuous extension of a rational rotation (see Proposition 2.3.1),7
2. In the purely affine case (i.e. when f = 0) for each α.
More precisely, in case (2) we have:
, : T2 → T2 be given by (x, y) -→
Theorem 1.0.5 ([23]) Let α ∈ R and c ∈ Z. Let T
8
, ⊥ µ.
(x + α, cx + y + γ ). Then T
Let us now describe how we check conditions (a), (b) and (c). Let α ∈
/ Q and let
T x = x + α. In either setting (purely affine or with a non-trivial perturbation) the
base rotation T r × T s is the same. Its ergodic components are obtained by taking the
2
partition of T2 into closed invariant sets Ac1 = Ar,s
c1 := {(x, y + c1 ) ∈ T : sx = r y},
1
c1 ∈ [0, r ). In fact, these sets are at the same time the minimal components of T r × T s
and they are uniquely ergodic. Thus, we are interested in the action of (Tc, f )r ×(Tc, f )s
6 We recall that under the continuity assumptions on f , even, it is open whether f is not a quasi-coboundary
for a generic set of α.
7 Liu and Sarnak [23] prove analogous result for smooth f .
8 This result was first proved in a more general context in [23] by different methods.
123
The Möbius function and continuous extensions of rotations
2
on the sets Ic1 = Icr,s
1 := A c1 × T . It turns out that this is equivalent to dealing with
extensions of T by the following T2 -valued cocycles:
.
ψc1 (x) = ψ (r ) (r x), ψ (s) (sx + c1 ) ,
where ψ(x) = f (x) + cx and c1 ∈ [0, r1 ). The cocycle ψc1 is ergodic if and only if
e2πi
#
Aψ (r ) (r x)+Bψ (s) (sx+c1 )
$
is not a multiplicative coboundary
for A, B ∈ Z, A2 + B 2 ̸= 0 (see Remark 2.5.10)9 . This is the situation we aim for
in course of the proof of Theorem 1.0.3. For a generic α we indeed obtain ergodicity
of ψc1 for all c1 —for the details see Corollary 2.5.8. Statement (b) follows from
the fact that we deal with compact group extensions of rotations. Finally, we show
that also (c) holds: given a non-trivial character χ ∈ +
T2 , we prove that for r and s
relatively prime and large enough, the corresponding integrals of χ ⊗ χ are zero. In
case of Theorem 1.0.5, the problem is in a sense more delicate—some of the sets Ic1
are too large to be the ergodic components and they need to be partitioned further into
smaller subsets. This refined partition will be however satisfying (a). Condition (b) is
proved in the same way as in Theorem 1.0.3. To prove that also (c) holds, we consider
again χ ∈ +
T2 .
For the convenience of the reader, the necessary facts concerning cocycles are
included in the “Appendix”.
2 Results
2.1 On the strategy of the proofs
Our approach to proving disjointness with the Möbius function was described in (a),
(b) and (c). We will now make some more comments on this method. Recall that in
view of Theorem 1.0.1, what we want to prove is that for a linearly dense set F of
g ∈ C(X ) and each x ∈ X , we have
"
g(T r n x)g(T sn x) = o(N ),
(7)
n≤N
for distinct, sufficiently large prime numbers r, s. When this is realized through (a),
(b) and (c), we prove more. Namely, for each g ∈ F and for sufficiently large primes
r ̸= s, the following holds for all x, y ∈ X :
"
g(T r n x)g(T sn y) = o(N )
(8)
n≤N
(the condition on r, s is independent of the choice of x and y).
9 We identify T with the multiplicative circle S1 = {z ∈ C : |z| = 1}. We will use both, the additive and
the multiplicative notation, whichever is more convenient for us at the moment. In particular, e2πi x is to be
understood multiplicatively and x in the exponent is to be understood additively.
123
J. Kułaga-Przymus, M. Lemańczyk
Remark 2.1.1 1. Let k ≥ 1 and suppose that, for sufficiently large primes r ̸= s, (8)
holds for T k and g ∈ C(X ), for all x, y ∈ X . Then
. #
" $ " - k rn r j . #
$
g T r (kn+ j) x g T s(kn+ j) x =
g (T ) T x g (T k )sn T s j x
n≤N
n≤N
= o(N )
for 0 ≤ j < k.
It follows that for sufficiently large primes r ̸= s, (7) holds for T and g ∈ C(X )
for all x ∈ X . In particular, in order to prove Sarnak’s conjecture for T , it suffices
to check conditions (a), (b) and (c) for T k for some k ≥ 1.
2. Notice that whenever the conditions (a), (b) and (c) are satisfied for some homeomorphism T then they are also satisfied for T −1 .
3. It is unclear how to prove directly that if T k for some k ∈ Z\{0} is disjoint with µ
then also T is disjoint from µ, or even to show that if the assumptions of the KBSZ
criterion are satisfied for T k for some k ∈ Z\{0} then they are satisfied for T .
Let now Tϕ : X × T → X × T be a continuous circle group extension of a homeomorphism T : X → X by ϕ : X → T.
Remark 2.1.2 Suppose that Tϕ ⊥ µ. Then for any k ≥ 1 also Tkϕ ⊥ µ as Tkϕ is a
topological factor of Tϕ .
Remark 2.1.3 In the case of affine extensions of rotations, Remark 2.1.1 (1) and
Remark 2.1.2 are complementary in the following sense. Let T (α) stand for the rotation
(α)
T (α) x = x + α and let ψ(x) = x. Then (Tψ k )k = T
(1), the following implication holds:
/
(α)
(a), (b) and (c) hold for T
kψ+ k−1
2 α
(α)
kψ+ k−1
2 α
0
and, by Remark 2.1.1
(α)
⇒ Tψ k ⊥ µ.
2.2 General remarks
From now on, our assumption will be that
r, s ≥ 3 are odd and relatively prime.
Let α ∈
/ Q and denote by T : T → T the irrational rotation T x = x +α. For c1 ∈ [0, r1 )
10
let
2
Ic1 = Icr,s
:= Ac1 × T2 , where Ac1 = Ar,s
c1 := {(x, y + c1 ) ∈ T : sx = r y}.
1
(9)
Lemma 2.2.1 The decomposition of T2 into minimal components of T r × T s consists
of sets Ac1 , c1 ∈ [0, r1 ). It is the same as the ergodic decomposition. Moreover,
(T r × T s )| Ac1 is topologically isomorphic to T . The isomorphism is given by
10 When r, s are fixed, we will write I and A . For r, s varying, we will use I r,s and Ar,s .
c1
c1
c1
c1
123
The Möbius function and continuous extensions of rotations
W = Wc1 : Ac1 → T, W (x, y + c1 ) = ax + by,
(10)
with a, b ∈ Z are such that ar + bs = 1.
Proof Notice first that the sets Ac1 are closed and invariant under T r × T s and
1
1
2
c1 ∈[0, r1 ) Ac1 = T . Fix c1 ∈ [0, r ), let a, b ∈ Z be such that ar + bs = 1 and
let W be given by (10). Then W ◦ (T r × T s )| Ac1 = T ◦ W . For (x, y + c1 ) ∈ Ac1 ,
we have r (ax + by) = x and s(ax + by) = y. Therefore, W is bijective. Moreover,
W preserves the measure, as rotations are uniquely ergodic.
2
⊓
For a measurable function ψ : T → T, let , : T2 → T2 be given by
.
,(x, y) = ψ (r ) (x), ψ (s) (y) .
(11)
Then clearly (Tψ )r × (Tψ )s is topologically isomorphic to (T r × T s ), .
Lemma 2.2.2 For c1 ∈ [0, r1 ) the sets Ic1 are invariant under (T r × T s ), .
Moreover, (T r × T s ), | Ic1 is topologically isomorphic to Tψc1 , where ψc1 (x) =
(ψ (r ) (r x), ψ (s) (sx + c1 )). The isomorphism is given by
V = Vc1 : Ic1 → T3 , V (x, y + c1 , u, v) = (ax + by, u, v),
(12)
with a, b ∈ Z such that ar + bs = 1.
Proof Fix c1 ∈ [0, r1 ), let a, b ∈ Z be such that ar + bs = 1 and let V be given
by (12). Then
and
V ◦ (T r × T s ), (x, y + c1 , u, v)
.
= V x + r α, y + sα + c1 , u + ψ (r ) (x), v + ψ (s) (y + c1 )
.
= ax + by + α, u + ψ (r ) (x), v + ψ (s) (y + c1 )
Tψc1 ◦ V (x, y + c1 , u, v) = Tψc1 (ax + by, u, v)
.
= ax + by + α, u + ψ (r ) (r (ax + by)), v + ψ (s) (s(ax + by) + c1 ) .
Moreover, r (ax + by) = x and s(ax + by) = y, which completes the proof.
2
⊓
Remark 2.2.3 The inverse of Vc1 : Ic1 → T3 is given by
Vc−1
(t, u, v) = (r t, st + c1 , u, v).
1
Moreover, ar t + bst = t for a, b ∈ Z such that ar + bs = 1.
123
J. Kułaga-Przymus, M. Lemańczyk
2.3 Continuous extensions of rational rotations
In this section we will prove the following:
Proposition 2.3.1 Let p, q ∈ N, let f : T → T be continuous and let R : T × T →
T × T be given by R(x, y) = (x + qp , f (x) + y). Then R ⊥ µ.
We will need an auxiliary lemma:
Lemma 2.3.2 Assume that S(x, y) = (x, f (x) + y) with f : T → T continuous.
22 we have
Then for each (xi , yi ) ∈ T2 , i = 1, 2, χ ∈ T
$
1 " # rn
χ S (x1 , y1 ) χ (S sn (x2 , y2 )) → 0 as N → ∞
N
(13)
n≤N
for sufficiently large prime numbers r ̸= s, whenever χ ̸= η ⊗ 1T (η ∈ +
T) and
/ Q or f (x2 ) ∈
/ Q.
f (x1 ) ∈
Proof Note that for each m ≥ 1,
S m (x, y) = (x, m f (x) + y).
Let χ (x, y) = e2πi(ax+by) for some a, b ∈ Z, with b ̸= 0 by assumption. Hence
$
1 " # rn
χ S (x1 , y1 ) χ (S sn (x2 , y2 ))
N
n≤N
= e2πi(a(x1 −x2 )+b(y1 −y2 ))
1 " 2πib(r f (x1 )−s f (x2 ))n
e
.
N
n≤N
If exactly one of the numbers f (x1 ) and f (x2 ) is irrational then the result follows
from Weyl’s criterion. If both f (x1 ) and f (x2 ) are irrational then there is at most one
pair (r, s) of relatively prime numbers such that r f (x1 ) − s f (x2 ) ∈ Q and we can
again make use of Weyl’s criterion, this time for r, s sufficiently large.
2
⊓
Remark 2.3.3 Notice that the above proof says more. Namely, the convergence in (13)
does not depend on y1 , y2 .
22 .
Proof (Proof of Proposition 2.3.1) We need to check (1) only for g = χ ∈ T
+
Suppose first that χ = η ⊗ 1T with η ∈ T. In this case, (1) follows from Sarnak’s
conjecture for finite systems.
T. Note that
Assume now that χ is not of the form η ⊗ 1T with η ∈ +
R q (x, y) = (x, f q (x) + y),
where f q (x) = f (x) + f (x + q1 ) + · · · + f (x +
q−1
q ).
(14)
Given n ≥ 1, we take n ′ such
22 , we have
that n = qn ′ + j with 0 ≤ j < q. Then, for each χ ∈ T
. #
$$
′
′ #
χ (R r n (x, y))χ (R sn (x, y)) = χ R qr n (R r j (x, y)) χ R qsn R s j (x, y) ,
123
The Möbius function and continuous extensions of rotations
where the first coordinates of the points R r j (x, y), R r j (x, y) belong to the finite set
{x, x + q1 , . . . , x + q−1
q } (hence do not depend on r, s). Hence, to obtain
$
1 " # rn
χ R (x, y) χ (R sn (x, y)) → 0
N
n≤N
for sufficiently large prime numbers r ̸= s, we need to show that
1
N
"
n ′ ≤N /q
.
′
′
χ R qr n (x1 , ∗) χ (R qsn (x2 , ∗)) → 0,
(15)
for x1 , x2 ∈ {x, x + 1/q, . . . , x + q−1
q }. It follows by Lemma 2.3.2, Remark 2.3.3
/ Q or f q (x2 ) ∈
/ Q. Suppose now that
and (14), that (15) holds, provided that f q (x1 ) ∈
f q (x + jr p/q), f q (x + jsp/q) ∈ Q for some 0 ≤ j < q. This is possible only if
f q (x) ∈ Q since f q (·) is constant on the orbit of x under the rotation x -→ x + qp .
Moreover, if n = qn ′ + j with 0 ≤ j < q then
f (n) (x) :=
n−1
"
i=0
′
f (x + i p/q) = f ( j) (x) + f (qn ) (x + j p/q) = f ( j) (x) + n ′ f q (x).
(16)
It follows that
1 "
χ (R n (x, y))µ(n)
N
n≤N
=
=
=
=
" 1
N
0≤ j<q
" 1
N
0≤ j<q
"
0≤ j<q
"
e
"
n ′ ≤N /q
"
#
$
#
$
χ x + (qn ′ + j p/q, f ( j) (x) + n ′ f q (x) + y)µ qn ′ + j
. #
$
χ x + j p/q, f ( j) (x) + n ′ f q (x) + y µ qn ′ + j
n ′ ≤N /q
#
$
2πi a(x+ j p/q)+b f ( j) (x)
e2πi(a(x+ j p/q)+b f
0≤ j<q
( j) (x))
1
N
1
N
"
n ′ ≤N /q
"
n ′ ≤N /q
′
e2πib fq (x)n µ(qn ′ + j)
c ′
e2πib d n µ(qn ′ + j),
where χ (x, y) = e2πi(ax+by) , f q (x) = dc with a, b, c, d ∈ Z, b ̸= 0 and d > 0. By
seting n ′ = dn ′′ + k with 0 ≤ k < d, we obtain qn ′ + j = qdn ′′ + (qk + j) and
rewriting the sum to the form
"
" 1
N
0≤ j<q 0≤k<d
"
n ′′ ≤N /(dq)
#
$
A j,k µ qdn ′′ + kq + j ,
(1) again follows from Sarnak’s conjecture for finite systems.
2
⊓
123
J. Kułaga-Przymus, M. Lemańczyk
2.4 Affine case
Recall that we are interested in the disjointness with the Möbius function of (x, y) -→
(x + α, cx + y + γ ), where c ∈ Z and α, γ ∈ R.
Remark 2.4.1 If c ̸= 0, it follows immediately from Remark 2.1.2 that instead of
(x, y) -→ (x + α, y + cx + γ ), we can consider (x, y) -→ (x + α, y + x + γc ). By
Proposition 2.3.1, this shows that it suffices to consider only the cases c = 1 and
c = 0, with α ∈
/ Q.
2.4.1 Case c = 1, α ∈
/Q
We will now deal with (x, y) -→ (x +α, y + x +γ ), i.e. with Tψ , where ψ(x) = x +γ .
Let , be given by (11), i.e.
/
0
(r − 1)r
(s − 1)s
,(x, y) = r x +
α + r γ , sy +
α + sγ .
2
2
Recall that given r, s ∈ N (odd and relatively prime) and c1 ∈ [0, r1 ), we have
3
4
Ic1 = Icr,s
= (x, y + c1 , u, v) ∈ T4 : sx = r y .
1
For c1 such that r s((s − r )γ − r c1 ) ∈ αQ + Q and c2 ∈ [0, r12 ), define
3
Jc1 ,c2 =Jcr,s
:=
(x, y + c1 , u, v + c2 ) ∈ T4 : sx = r y and
1 ,c2
0
5
/
r −s
l0 s 2 u = l0 r 2 v + l0 r s
− k0 (ax + by) ,
2
where l0 = l0r,s,c1 is the smallest positive integer such that
l0 r s((s − r )γ − r c1 ) ∈ αZ + Z
(17)
and k0 = k0r,s,c1 ∈ Z is such that
l0 r s((s − r )γ − r c1 ) − k0 α ∈ Z.
(18)
Lemma 2.4.2 For c1 ∈ [0, r1 ) the homeomorphism Tψc1 : T3 → T3 , where ψc1 (x) =
(ψ (r ) (r x), ψ (s) (sx + c1 )), is topologically isomorphic to Tϕc1 : T3 → T3 , where
ϕc1 : T → T2 is given by ϕc1 (x) = (r 2 x + r γ , s 2 x + sc1 + sγ ). The isomorphism is
given by
/
0
r (r − 1)
s(s − 1)
U = Uc1 : T3 → T3 , U (x, u, v) = x, u −
x, v −
x .
2
2
123
(19)
The Möbius function and continuous extensions of rotations
Proof We have
.
ψc1 (x) = ψ (r ) (r x), ψ (s) (sx + c1 )
0
/
r (r − 1)
s(s − 1)
α + r γ , s(sx + c1 ) +
α + sγ .
= r2x +
2
2
.
Since for θ (x) = − r (r2−1) x, − s(s−1)
x
we have
2
/
0
r (r − 1) s(s − 1)
α,
α = θ (x) − θ (x + α),
2
2
it follows that U given by (19) is indeed the required isomorphism. Notice that θ is
continuous, whence U is a homeomorphism.
2
⊓
Proposition 2.4.3 The decomposition of T4 into minimal components for (T r × T s ),
is the same as the decomposition into ergodic components. It consists of sets of the
/ αQ + Q and Jc1 ,c2 for c1 ∈ αQ + Q, where c1 ∈ [0, r1 ), c2 ∈ [0, r12 ).
form Ic1 for c1 ∈
Moreover, on each such component, (T r × T s ), is uniquely ergodic. In particular,
each point in T4 is generic (for the relevant invariant measure).
Proof In view of Lemma 2.4.2, instead of (T r × T s ), , we may consider Tϕc1 , with
ϕc1 (x) = (r 2 x + r γ , s 2 + sc1 + sγ ).
We will show first that the decomposition into minimal components for (T r ×
s
T ), is the same as the decomposition into ergodic components. By Remark 2.5.15,
Proposition 2.5.9 and Proposition 2.5.12, it suffices to show that the equation
χ ◦ ϕc 1 = ξ − ξ ◦ T
(20)
has a measurable solution ξ : T → T for χ (u, v) = Au + Bv for A, B ∈ Z with
A2 + B 2 ̸= 0 if and only if it has a continuous one. We have
.
χ ◦ ϕc1 (x) = χ r 2 x + r γ , s 2 x + sc1 + sγ
.
= Ar 2 + Bs 2 x + (Ar + Bs)γ + Bsc1 .
By [3], χ ◦ ϕc1 can be a measurable coboundary only if
Ar 2 + Bs 2 = 0
and
(Ar + Bs)γ + Bsc1 ∈ αZ + Z.
(21)
Then the solution to (20) is given by ξ(x) = −kx, where k ∈ Z is such that (Ar +
Bs)γ + Bsc1 − kα ∈ Z. In particular, all measurable solutions to (20) are continuous.
We will now describe the ergodic (i.e. minimal) components of each Tϕc1 and show
that they are uniquely ergodic. Suppose that χ ◦ ϕc1 is a coboundary, i.e. for some
measurable ξ , we have (20). It follows from (21) (recall that r ̸= s are coprime) that
123
J. Kułaga-Przymus, M. Lemańczyk
r 2 |B and s 2 |A, and therefore A = A′ s 2 and B = −A′r 2 for some A′ ∈ Z. Hence, the
second part of condition (21) takes the form
A′r s((s − r )γ − r c1 ) ∈ αZ + Z.
(22)
Having this in mind, we consider two cases:
/ Qα + Q,
1. (s − r )γ − r c1 ∈
2. (s − r )γ − r c1 ∈ Qα + Q.
In case (1), it follows immediately from the first part of the proof that (T r × T s ), | Ic1
is ergodic and minimal. The unique ergodicity of (T r × T s ), | Ic1 follows from Proposition 2.5.16. Consider now case (2). We will describe characters χ , such that (20) has
a (measurable and continuous) solution. In view of and (17) and (22), A′ = al0 for
some a ∈ Z. Therefore, a measurable solution ξ : T → T to (20) exists precisely for
the characters χ of T2 of the form
χ (u, v) = al0 s 2 u − al0 r 2 v
for a ∈ Z.
(23)
Denote the set of such characters by 0 = 0top . It is easy to see that
3
4
ann 0 = (u, v) ∈ T2 : l0 s 2 u = l0 r 2 v .
We claim that
3
4
3
2
2
:=
(t,
u,
v
+
c
)
∈
T
:
l
s
u
=
l
r
v
−
k
t
J,cr,s
2
0
0
0
,c
1 2
for c2 ∈ [0, 1/r 2 )
are minimal components of Tϕc1 . Indeed:
• Each J,cr,s
1 ,c2 is closed,
• By (18), each J,cr,s
1 ,c2 is Tϕc1 -invariant,
1
r,s
3
,
• c2 Jc1 ,c2 = T (notice that the projection of J,cr,s
1 ,c2 onto the first two coordinates
is equal to T2 ),
,r,s
• (u, v) ∈ ann 0 if and only if J,cr,s
1 ,c2 + (0, u, v) = Jc1 ,c2
(see Proposition 2.5.12). Unique ergodicity follows, as in the previous case, from
Proposition 2.5.16.
To find the minimal component corresponding to each of the sets J,cr,s
1 ,c2 described
via
U
◦
V
.
We
have
above, we need to find the preimage of J,cr,s
c
c
,c
1
1
1 2
6
/
0
/
0
5
# r,s $
,c ,c = (t, u, v + c2 ) : l0 s 2 u − r (r − 1) t = l0 r 2 v − s(s − 1) t − k0 t
J
Uc−1
1
1 2
2
2
/
0 5
6
r −s
− k0 t ,
= (t, u, v + c2 ) : l0 s 2 u = l0 r 2 v + l0 r s
2
123
The Möbius function and continuous extensions of rotations
whence, by Remark 2.2.3, 6
) = (r t, st + c1 , u, v + c2 ):
(Uc1 Vc1 )−1 ( J,cr,s
1 ,c2
/
0 5
r −s
l0 s u = l0 r v + l0 r s
− k0 t
2
3
= (x, y + c1 , u, v + c2 ) : sx = r y and
0
5
/
r −s
2
2
l0 s u = l0 r v + l0 r s
− k0 (ax + by) .
2
2
2
#
$
r,s
Therefore (Uc1 Vc1 )−1 J,cr,s
1 ,c2 = Jc1 ,c2 , which completes the proof.
2
⊓
r,s
4
Remark 2.4.4 The sets Icr,s
1 are translates of I0 , which is a subgroup of T . Therefore,
r
the conditional measures given by the ergodic decomposition of (T × T s ), , which
/ Qα + Q, are translates
are supported on the sets Ic1 for c1 such that (s − r )γ − r c1 ∈
of Haar measure on I0r,s .
,r,s
For c1 with (s − r )γ − r c1 ∈ Qα + Q, the sets J,cr,s
1 ,c2 are translates of Jc1 ,0 ,
3
which is a subgroup of T . As before, the conditional measures given by the ergodic
. Notice that Vc1 carries
decomposition of Tϕc1, are translates of Haar measure on J,cr,s
1 ,0
a translate of Haar measure to Haar measure. Therefore, also the conditional measures
given by the ergodic decomposition of (T r × T s ), , which are supported on the sets
Jcr,s
1 ,c2 , are translates of the corresponding Haar measures.
+ and let H = ker χ . Then
Remark 2.4.5 Let G be a compact Abelian group. Let χ ∈ G
χ is constant on each coset of H . Moreover, if x + H ̸= y + H then χ (x + H ) ̸=
χ (y + H ).
+ and let H ⊂ G be a subgroup. Then the integral of χ with
Remark 2.4.6 Let χ ∈ G
respect to Haar measure on each coset of H is zero if and only if χ | H is not constant.
+ Then there exists
Lemma 2.4.7 Let G be a compact metric Abelian group. Let χ ∈ G.
δ > 0 such that whenever {y1 , . . . , yn } ⊂ G is a δ-net and χ (yi ) = 1 for i = 1, . . . , n
then χ ≡ 1.
Proof Let d stand for the metric on G. Since χ is uniformly continuous, there exists
δ > 0 such that d(x, y) < δ implies |χ (x) − χ (y)| < 41 . Since {y1 , . . . , yn } is a δ-net,
it follows that |χ (G) − 1| < 21 . This is however possible only when χ ≡ 1, as χ (G)
is a closed subgroup of S1 .
2
⊓
Lemma 2.4.8 Let χ ∈ +
T4 be a non-trivial character. If r, s ∈ N are large enough
then χ | Ic1 is not constant for c1 ∈ [0, r1 ).
Proof Fix 1 ̸≡ χ ∈ +
T4 and let δ > 0 be as in Lemma 2.4.7. In view of Lemma 2.4.7,
we need to show that whenever r, s are large enough then there exists a δ-net of T4
r,s
r,s
1
contained in Icr,s
1 for c1 ∈ [0, r ). Since Ic1 is a translate of I0 and the third and fourth
coordinate in I0 is arbitrary, it suffices to prove that for r, s sufficiently large we can
r,s
the proof it suffices to
always find a δ-net of T2 contained
. in the set A0 . To complete
4
3consider sets of the form
i j
s, r
: 0 ≤ i < s, 0 ≤ j < r .
2
⊓
123
J. Kułaga-Przymus, M. Lemańczyk
Lemma 2.4.9 Let χ ∈ +
T4 be a non-trivial character. If r, s ∈ N are large enough
then χ | Jcr,s,c is not constant for c1 ∈ [0, r1 ) such that c1 ∈ αQ + Q and c2 ∈ [0, r12 ).
1 2
Proof Fix 1 ̸≡ χ ∈ +
T4 and let δ > 0 be as in Lemma 2.4.7. Since Jcr,s
1 ,c2 is a translation
r,s
of Jc1 ,0 , by Lemma 2.4.7, it suffices to prove that for r, s sufficiently large there exists
a δ-net of T4 in Jcr,s
.
1 ,0
onto the first two coordinates is equal to Ar,s
Notice first that the projection of Jcr,s
c1 .
1 ,0
2
Indeed, for any z ∈ T the equation l0 s u = l0 r 2 v + z has a solution (u, v) as T is an
infinitely divisible group. By the proof of the previous lemma, we can find a δ-net of
T2
7
8
(xi , y j ) : 0 ≤ i, j < n ⊂ Ar,s
c1 .
Moreover, by the infinite divisibility of T, for each (i, j) there exist u i, j , vi, j such that
. Moreover, for 0 ≤ a, b < l0 s 2
(xi , y j , u i, j , vi, j ) ∈ Jcr,s
1 ,0
/
xi , y j , u i, j +
a
b
, vi, j +
2
l0 s
l0 s 2
0
.
∈ Jcr,s
1 ,0
Therefore, whenever r, s are sufficiently large, the set
6/
xi , y j , u i, j
a
b
+
, vi, j +
l0 s 2
l0 s 2
0
: 0 ≤ i, j < n, 0 ≤ a, b < l0 s
is the required δ-net.11 This completes the proof.
2
5
2
⊓
Proposition 2.4.10 For*any 1 ̸≡ χ ∈ +
T4 whenever r, s ∈ N are odd, relatively prime
and large enough then I χ dλ I = 0 for each ergodic component of (T r × T s ), ,
where λ I is the relevant invariant measure.
Proof The assertion follows immediately from Proposition 2.4.3, Lemmas 2.4.8 and
2.4.9.
2
⊓
Proof of Theorem 1.0.5 in case c = 1, α ∈
/ Q. We only need to check that (a),
(b), (c) hold. Conditions (a) and (b) follow from Proposition 2.4.3. Moreover, by
Proposition 2.4.10, also condition (c) is satisfied (we take F = +
T2 ).
2
⊓
2.4.2 Case c = 0, α ∈
/Q
Proof of Theorem 1.0.5 in case c = 0, α ∈
/ Q. We have Tψ (x, u) = (x + α, u + γ ), i.e.
Tψ is a rotation on T2 . The decomposition into minimal components of (T r × T s ),
consists of the cosets of I0 = I0r,s = {(x, y, 0, 0) ∈ T4 : sx = r y}. They are at the
same the ergodic components which are moreover uniquely ergodic, and we conclude
as previously.
2
⊓
11 Notice that since l ≥ 1, the condition on r, s is independent of c .
0
1
123
The Möbius function and continuous extensions of rotations
2.5 Generic case: compact group extensions
Let f : R → R (periodic of period 1) be in L 2 (T). Denote by
f (x) =
"
n∈Z
+
f (n)e2πinx
the Fourier expansion of f . Recall that our goal is to prove disjointness of
T2 ∋ (x, y) -→ (x + α, y + cx + f (x)) ∈ T2 ,
with the Möbius function µ for a generic set of α (under some additional assumptions
on f ).
Recall the following result.
!
f (n)e2πinx is in C 1+δ (T)
Theorem 2.5.1 ([18,21])12 Suppose that f (x) = n∈Z +
for some δ > 0, and has zero mean. Denote by T : T → T the irrational rotation
x -→ x + α. Assume that for a sequence ( pn /qn )n∈N of rational numbers we have
and
9
9
9+
f (qn )9
9 >c>0
9
!
9
9+
k≥1 f (kqn )
9
9
9
p 9
9α − qnn 9 qn
9 → 0.
9
9+
f (qn )9
(24)
(25)
Then, for each λ ∈ S1 , the cocycle λe2πi f (·) is not a T -coboundary.
We will now prove a modified version of the above theorem. It will be our main
tool in course of the proof of Theorem 1.0.3.
!
f (n)e2πinx is in C 1+δ (T) for some δ >
Theorem 2.5.2 Suppose that f (x) = n∈Z +
0, and has zero mean. Denote by T : T → T the irrational rotation x -→ x + α. Let
r, s ∈ N be relatively prime. Assume that ( pn /qn )n∈N is a subsequence of convergents
of α in its continued fraction expansion such that
9
9
9+
f (qn )9
9 > c0 > 0,
9
!
9
9+
k≥1 f (kqn )
9
9
9+
f (qn )9
9 !
9 > c0 > 0, whenever s|qn ,
9
9
9+
f (qn )9 + k≥1 9 +
f (k rs qn )9
(26)
(27)
12 In [18] f is assumed to be of class C 2 and the proof, in fact, requires that ! n| +
f (n)| < ∞. See [21]
for the proof of Theorem 2.5.1 with f ∈ C 1+δ (T) for some δ > 0.
123
J. Kułaga-Przymus, M. Lemańczyk
and
9
9
9
p 9
9α − qnn 9 qn
9 → 0.
9
9+
f (qn )9
(28)
Then for each λ ∈ S1 , h ∈ R and A, B ∈ R with A2 + B 2 ̸= 0 the cocycle
(r )
(s)
λe2πi(A f (r x)+B f (sx+h)) is not a T -coboundary.
Proof Fix A, B ∈ R with A2 + B 2 ̸= 0, relatively prime numbers r, s and h ∈ R. Set
F(x) := A f (r ) (r x) + B f (s) (sx + h).
For any k ∈ N we have
f (k) (x) =
"
n∈Z
Therefore,
F(x) = A
"
n∈Z
2πinkα
1−e
+
e2πinx .
f (n)
1 − e2πinα
2πinr α
1−e
+
e2πinr x + B
f (n)
1 − e2πinα
"
n∈Z
2πinsα
1−e
+
e2πin(sx+h) . (29)
f (n)
1 − e2πinα
Suppose first that A · B = 0. We may assume without loss of generality that A = 0
and B ̸= 0. Then, for all n ∈ N, by (29), we have
2πinsα
Therefore, since
and
1−e
+
e2πinh .
F(sn)
= B+
f (n)
1 − e2πinα
9
9
9 1 − e2πiqn sα 9
9
9
9 1 − e2πiqn α 9 → s as n → ∞,
9
9
9
9
9 F(sq
+ n )9 ≥ |B| · s · 9 +
f (qn )9 for n sufficiently large
2
9
9
9
9
9
9
9 9 1 − e2πimsqn α 9
9
9+
9
9
9 F(msq
9
9
9
9
+
+
=
|B|
·
f
(mq
·
)
)
n
n
9 1 − e2πimqn α 9 ≤ |B| · s · f (mqn )
(30)
(31)
(32)
for all n ∈ N, m ∈ Z. It follows immediately by (31) and (32) that for n sufficiently
large
9
9
9
9
9
9
9 F(sq
9+
+ n )9
|B| · 2s · 9 +
f (qn )9
f (qn )9
9≥!
9
9= !
9
9
9
!
9
9+
9
9+
2 m≥1 9 +
f (mqn )9
m≥1 F(msqn )
m≥1 |B| · s · f (mqn )
123
9
9
9
sp 9
9α − sqnn 9 sqn
2
9 ≤
9
9 F(sq
+ n )9
|B|
9
9
9
p 9
9α − qnn 9 qn
9 .
· 9
9+
f (qn )9
(33)
(34)
The Möbius function and continuous extensions of rotations
Now, (33) and (34) and the assumptions (26) and (28) imply that for n sufficiently
large
9
9
9 F(sq
+ n )9
c
9 > 0 >0
9
!
9
9 F(msq
+
2
n)
m≥1
9
9
9
sp 9
9α − sqnn 9 sqn
9 → 0.
9
9 F(sq
+ n )9
and
In view of Theorem 2.5.1, this completes the proof in case A · B = 0.
Suppose now that A · B ̸= 0. Applying the fact that for any absolutely summable
sequence (yn )n∈Z
"
n∈Z
13 to
"
yn =
r |n
yn +
"
r !n
yn =
"
ynr +
n∈Z
"
r !n
yn =
"
s|n
yn rs +
"
yn
r !n
the second summand in formula (29), we obtain
F(x) =
"/
s|n
+A
:
0
- r . 1 − e2πinr α
1 − e2πinr α
2πin rs h
+
+
B
f
n
e
e2πinr x
r
1 − e2πinα
s 1 − e2πin s α
;<
=
A+
f (n)
"
s !n
ar n
+
f (n)
1 − e2πir nα
1 − e2πinα
e2πinr x + B
"
r !n
2πinsα
1−e
+
e2πinh e2πinsx .
f (n)
1 − e2πinα
(35)
For m ∈ N, let Bm := {n ∈ N : m ! qn }. We will consider the following cases:
1. Br or Bs is infinite,
2. Both sets Br and Bs are finite.
We will cover first case (1). Without loss of generality we may assume that Br is
infinite. Suppose that
λe2πi
#
A f (r ) (r x)+B f (s) (sx+h)
$
is a coboundary.
(36)
Since f (r ) (r x) is r1 -periodic, it follows immediately that
λe2πi
#
A f (r ) (r x)+B f (s) (sx+ rs +h)
$
is also a coboundary.
(37)
Hence, dividing the expression from (36) by the one from (37), we conclude that
e2πi B
#
f (s) (sx)− f (s) (sx+ rs )
$
is a coboundary as well.
(38)
13 These equalities hold for all r, s ∈ N.
123
J. Kułaga-Przymus, M. Lemańczyk
We will show that this is impossible in view of Theorem 2.5.1. Indeed, we have
.
s . " + 1 − e2πinsα 2πin rs
=
1
−
e
e2πinsx .
g(x) := f (s) (sx) − f (s) sx +
f (n)
r
1 − e2πinα
n∈Z
We claim that for n ∈ Br ,
!
and
|+
g (sqn )|
> c1
g (ksqn )|
k≥1 |+
Indeed, for n ∈ Br , we have
9
9
9α −
9
spn 9
sqn 9 sqn
|+
g (sqn )|
for some c1 > 0
(39)
→ 0.
(40)
9
9
9
9
9 9 1 − e2πisqn α 9 99 2πiq s
n r − 19 .
9 · 9e
|+
g (sqn )| = 9 +
f (qn )9 · 99
9
1 − e2πiqn α 9
(41)
Since r ! qn and r and s are relatively prime,
9
9 9
9
1
9 2πiqn rs
9 9
9
− 19 ≥ 9e2πi r − 19 =: c2 > 0.
9e
(42)
Using again (30), we obtain from (41) and (42) that for n sufficiently large,
|+
g (sqn )| ≥
c2 s 99 + 99
· f (qn ) .
2
(43)
9 2πiksq α 9
n 9
9
On the other hand, since 9 1−e
9 ≤ s,
1−e2πikqn α
"
k≥1
|+
g (ksqn )| =
9
9
9
"9
9 9 1 − e2πiksqn α 9 99 2πiq s
n r − 19
9 · 9e
9+
f (kqn )9 · 99
9
9
1 − e2πikqn α
k≥1
9
9
"9
"9
9
9
s
9
9
9+
9+
≤
f (kqn )9 · s · 9e2πiqn r − 19 ≤ 2s
f (kqn )9 .
k≥1
(44)
k≥1
Thus, using (43), (44) and (26) we obtain, for n large enough,
9
9
f (qn )9
c2 s 9 +
|+
g (sqn )|
c c
9 > 0 2,
9
!
≥
!
9
9
+
g (ksqn )|
4
4s k≥1 f (kqn )
k≥1 |+
which shows that (39) holds. Using (43) and (28), we obtain that (40) also holds:
9
9
9
9
9
9
9
9
9
9
sp 9
pn 9
α
−
2 9α − qpnn 9 sqn
9α − sqnn 9 sqn
9
qn 9 qn
2
9
9 =
9 → 0.
· 9
≤
9+
|+
g (sqn )|
c2
c2 s 9 +
f (qn )9
f (qn )9
123
The Möbius function and continuous extensions of rotations
(s)
(s)
s
It follows from Theorem 2.5.1 that λe2πi B( f (sx)− f (sx+ r )) cannot be a coboundary,
contrary to (38). This completes the proof in case (1).
We cover now case (2). We claim that for n ∈
/ Bs (i.e. n such that s|qn ), for some
c3 > 0, we have
|arqn |
!
> c3 > 0
(45)
m≥1 |amrqn |
and
|α −
r pn
rqn |rqn
|arqn |
→ 0.
(46)
!
To prove (45), we will estimate |arqn | from below and m≥1 |amrqn | from above in
an appropriate way. We begin by estimating |arqn |. We have
9
9
9 - r .9 99 1 − e2πiqn r α 99
9 1 − e2πirqn α 9
9
9
9
9
9
+
f qn 9 · 99
− |B| · 9 +
|arqn | ≥ | A| · | f (qn )| · 9
r
9
1 − e2πiqn α 9
s
1 − e2πiqn s α
9
9
9
- r .9 9 1 − e2πiqn r α 9
r
9
9
9
f qn 9 · 99
(47)
≥ |A| · | +
f (qn )| · − |B| · 9 +
r 9,
2
s
1 − e2πiqn s α
where the latter inequality follows from (30) and is valid for n sufficiently large. It
follows by (27) that for n ∈
/ Bs ,
9 - r .9
1
9
9+
f (qn )|.
9 f qn 9 < | +
s
c0
(48)
r
We will now estimate |1 − e2πiqn s α |. Notice that
9 2πi x
9
9e
− e2πi y 9
2
<
<1
π
|x − y|
(49)
for x, y ∈ R such that 0 < |x − y| < 1. Since |1 − e2πiqn α | → 0, for n sufficiently
large,
9
9
9
19
s 99
9
9
91 − e2πi s 9 .
91 − e2πiqn α 9 <
πr
Therefore and by (49), for such n, we have
9
9
9 99 [q α]
1 9
1 9 1
n
9 2πi [qns α]
− qn α 99 = · |[qn α] − qn α|
− e2πiqn s α 9 < 99
9e
s
s
s
9
9
π 99
π 99 2πi[qn α]
9
9
<
· 9e
· 91 − e2πiqn α 9
− e2πiqn α 9 =
2s
2s
9
9
19
19
1 99
π
s 99
<
·
· 91 − e2πi s 9 =
· 91 − e2πi s 9 .
2s πr
2r
(50)
Since for all x, y ∈ R and all k ∈ N
9
9
9
9
9 2πikx
9
9
9
− e2πiky 9 ≤ k 9e2πi x − e2πi y 9 ,
9e
123
J. Kułaga-Przymus, M. Lemańczyk
it follows from (50) that
9
9 19
9
19
r 9
9 2πi [qnsα]r
9
− e2πiqn s α 9 < 91 − e2πi s 9 .
9e
2
(51)
Since qn is a denominator of α and s|qn , we have |c − qsn α| > | pn − qn α| for all
0 ≤ c ≤ qsn (see (56) in Sect. 1). Using this inequality and (49), we obtain
9 9
9
9
qn 9
qn 99
π 99
9
9
9
91 − e2πi s α 9 > 9c − α 9 > | pn − qn α| > 91 − e2πiqn α 9 .
2
s
(52)
By the first two lines of (50) and by (52), we have
9
9
9
9
qn 9
1 9
π 99
π π 99
9 2πi [qns α]
9
− e2πiqn s α 9 <
· 91 − e2πi s α 9 .
9e
91 − e2πiqn α 9 <
2s
2s 2
(53)
π π
Suppose that s|[qn α]. Then (53) implies 1 < 2s
· 2 , i.e. s < (π/2)2 . This is however
impossible, as s ≥ 3. Therefore s ! [qn α], which implies s ! [qn α]r , whence
9
9 9
9
[qn α]r 9
19
9
9
91 − e2πi s 9 ≥ 91 − e2πi s 9 .
(54)
Therefore, by (51) and (54),
9
9
9
9 9
9
[qn α]r 9
[qn α]r
r 9
r 9
9
9
9
91 − e2πiqn s α 9 ≥ 91 − e2πi s 9 − 9e2πi s − e2πiqn s α 9
9
9 19
9 19
9
[qn α]r 9
19
19
9
9
9
≥ 91 − e2πi s 9 − 91 − e2πi s 9 ≥ 91 − e2πi s 9 =: c4 > 0.
2
2
It follows that
9
9
9 1 − e2πirqn α 9
1
r
9
9
|1 − e2πirqn α | ≤ |1 − e2πiqn α | → 0.
r 9 ≤
9
c4
c4
1 − e2πiqn s α
This and (47), (48) imply that for n sufficiently large
f (qn )|
|arqn | ≥ c5 | +
We will estimate now
"
m≥1
Hence
|+
f (mqn )| <
!
1 +
| f (qn )|
c0
"
m≥1
123
m≥1 |amrqn |.
|amrqn | ≤
for some c5 > 0.
By (26) and (27) we have
and
" 99 1
r .99
f mqn 9 < | +
f (qn )|.
9+
s
c0
m≥1
|A|r + |B|s +
· | f (qn )|.
c0
(55)
The Möbius function and continuous extensions of rotations
Using this estimate and (55), we obtain
9
9
9arq 9
n
9
9≥
!
9amrq 9
m≥1
n
9
9
c5 9 +
f (qn )9
c0 c5
>0
9=
9
|A|r +|B|s 9 +
9
|A|r
+ |B|s
f (qn )
c0
and (45) follows. Notice that (55) together with (28) implies that also (46) is true. By
Theorem 2.5.1, we conclude that λe2πi F(x) cannot be a coboundary, which completes
the proof in case (2).
2
⊓
′
f (n) = o(1/n 1+δ )
Remark 2.5.3 Recall that if f ∈ C 1+δ (T) for some δ > 0 then +
for 0 < δ ′ < δ (the speed of convergence to zero depends on δ ′ ).
Lemma 2.5.4 (cf. Lemma 4 in [21]) Let g : N → (0, ∞) be a non-increasing
positive
!
function such that g(mn) ≤ g(m)g(n) for all m, n ∈ N and m≥1 g(m) < ∞. Let
(xn )n∈N ⊂ [0, ∞) be a summable sequence such that xn = o(g(n)), n ∈ N. Let
(xn k )k∈N be a subsequence of (xn )n∈N such that xn k > 0. Let b ≥ 1 and let
εk =
xnk +
Then εk ̸→ 0.
xn
! k
m≥1 x m[bn k ]
.
Proof We will choose a subsequence of (εk )k∈N recursively. Let C :=
Let k1 ≥ 1 and δ1 > 0 be such that
x n k1
g(n k1 )
> δ1 and
xn
≤ δ1
g(n)
!
m≥1
g(m).
for n > n k1 .
Suppose first that [bn k1 ] = n k1 . Then
x n k1 +
"
m≥1
xm[bn k1 ] = 2xn k1 +
≤ 2xn k1 + g
"
m≥2
xm[bn k1 ] ≤ 2xn k1 +
"
m≥2
g(m[bn k1 ])δ1
?$ "
#>
bn k1 δ1
g(m) ≤ 2xn k1 + g(n k1 )δ1 C < (C + 2)xn k1 .
m≥2
Suppose now that [bn k1 ] > n k1 . In a similar way as before, we obtain
x n k1 +
"
m≥1
xm[bn k1 ] ≤ xn k1 + g(n k1 )δ1
"
m≥1
g(m) < (C + 1)xn k1 .
Once we have chosen k1 , . . . , k j , we pick k j+1 > k j and δ j+1 > 0 such that
xn k j+1
g(n k j+1 )
> δ j+1 and
xn
≤ δ j+1
g(n)
for n > n k j+1 .
123
J. Kułaga-Przymus, M. Lemańczyk
As before, we obtain
"
m≥1
xm[bn k j+1 ] < (C + 2)xn k j+1
2
⊓
which completes the proof.
Remark 2.5.5 Given b1 , . . . , bl ≥ 1, under the assumptions of the above lemma, by a
diagonalizing procedure we can find an increasing sequence (n k )k∈N ⊂ N such that
εn k (bi ) > c > 0 for1 ≤ i ≤ l.
Corollary 2.5.6 Suppose that f ∈ C 1+δ (T) for some δ > 0 and f is not a trigonometric polynomial. Then for a generic α and any A, B ∈ R with A2 + B 2 ̸= 0,
any relatively prime numbers r, s, any h ∈ R and any λ ∈ T, the cocycle
(r )
(s)
λe2πi(A f (r x)+B f (sx+h)) is not a T -coboundary for T x = x + α.
Proof Notice first that if
λe2πi
#
A f (r ) (r x)+B f (s) (sx+h)
$
is a T -coboundary for T x = x + α
then also
λe
.
(s) (sx)
2πi A f (r ) (r x− hr
s )+B f
is a T -coboundary for T x = x + α.
Therefore, we may assume without loss of generality that r > s. Let (qn )n∈N be such
that +
f (qn ) ̸= 0. Then by (57), for a residual set of irrationals α, we have
9
9
9α −
9
pn 9
qn 9 qn
|+
f (qn )|
→0
along some subsequence of (qn )n∈N which, for convenience, we will still denote by
(qn )n∈N . In other words, (28) holds. It follows from Remark 2.5.3, Lemma 2.5.4 and
Remark 2.5.5 (for l = 2, b1 = 1, b2 = rs ) that (26) and (27) also hold. Therefore, we
can apply Theorem 2.5.2 to complete the proof.
2
⊓
Remark 2.5.7 Let c ̸= 0, r, s, A, B ∈ Z be such that Ar 2 + Bs 2 ̸= 0. Let ϕ(x) =
cx + f (x), where f : R → R is of class C 1+δ for some δ > 0 and periodic of period
(r )
(s)
1. Since the topological degree of e2πi(Aϕ (r ·)+Bϕ (s·+h)) is equal to (Ar 2 + Bs 2 )c,
(r )
(s)
an immediate consequence of (60) is that the cocycle e2πi(Aϕ (r ·)+Bϕ (s·+h)) is not
a coboundary for all h ∈ R. On the other hand, if Ar 2 + Bs 2 = 0, then for some λ of
modulus 1, we have
e2πi
123
#
Aϕ (r ) (r ·)+Bϕ (s) (s·+h)
$
= λ · e2πi
#
A f (r ) (r ·)+B f (s) (s·+h)
$
.
The Möbius function and continuous extensions of rotations
Corollary 2.5.8 Let c ∈ Z and suppose that f ∈ C 1+δ (T) for some δ > 0 and f
is not a trigonometric polynomial. Let ϕ(x) = cx + f (x). Then for a generic α the
(r )
(s)
cocycle (e2πiϕ (r ·) , e2πiϕ (s·+h) ) is ergodic (as a cocycle over T x = x + α) for all
relatively prime numbers r ̸= s and any h ∈ R.
Proof In view of Remark 2.5.10, the assertion follows immediately from Corollary 2.5.6 and Remark 2.5.7.
2
⊓
Proof of Theorem 1.0.3 Recall that Tc, f (x, y) = (x + α, y + cx + f (x)). We can
assume that f is not a trigonometric polynomial. Indeed, otherwise f is a coboundary
with the transfer function also being a trigonometric polynomial and the problem is
reduced to the affine case.
It follows from Lemma 2.2.2 and Corollary 2.5.8 that for a generic α the decomposition of T4 into minimal components of (Tc, f )r × (Tc, f )s is the same as the
decomposition into ergodic components: it consists of sets Ic1 , c1 ∈ [0, r1 ) (see (9)
on page 9). Moreover, each ergodic component is uniquely ergodic. Therefore, all
points are generic for (Tc, f )r × (Tc, f )s (for the relevant invariant measures). Thus,
conditions (a) and (b) are satisfied.
Finally, notice that by Lemma (2.4.8), F := +
T2 satisfies condition (c). This completes the proof.
2
⊓
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
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Appendix
Notation and basic facts
Automorphisms of standard Borel spaces Let (X, B, µ) be a standard probability Borel
space. By Aut(X, B, µ) we denote the space of all bi-measurable measure-preserving
bijections of X which we call automorphisms.
Irrational rotation We identify the multiplicative circle S1 = {z ∈ C : |z| = 1}
and T = R/Z with X = [0, 1) with addition mod 1. Therefore, real functions defined
on the circle are identified with one-periodic functions defined on R. Let λT denote
Lebesgue measure on X .
Assume that T : X → X is an irrational rotation, T x = x + α (mod 1), x ∈ X .
Clearly T ∈ Aut(X, B(X ), λT ). Let α = [0; a1 , a2 , . . .] be the continued fraction
expansion of α. Let
q0 = 1, q1 = a1 , qn+1 = an+1 qn + qn−1 ,
p0 = 0, p1 = 1, pn+1 = an+1 pn + pn−1
for n ≥ 1. The rationals pn /qn are called the convergents of α.
123
J. Kułaga-Przymus, M. Lemańczyk
Recall (see e.g. [20]) that every convergent pn /qn is a best approximation of α in
the following sense:
if
pn
c
̸=
and 0 < d ≤ qn then |c − dα| > | pn − qn α|.
d
qn
(56)
Recall also (see e.g. [21]) that given an infinite set {qn : n ∈ N} ⊂ N and a positive
real valued function R = R(qn ), the set
9
9
6
9
pn 99
9
< R(qn ),
A = α ∈ [0, 1) : for infinitely many n we have 9α −
qn 9
5
pn
are convergents of α is residual in T.
(57)
where
qn
Cocycles and group extensions Let T ∈ Aut(X, B, µ). For a locally compact second
countable Abelian group G 14 with a Haar measure λG , a measurable map ϕ : Z× X →
G is called a cocycle if
ϕ (n+m) (x) = ϕ (n) (x)ϕ (m) (T n x)
for each n, m ∈ Z.
The generator ϕ(x) = ϕ (1) (x) determines ϕ (n) (x) for any (n, x) ∈ Z × X :
⎧
# n−1 $
⎪
x ,
⎨ϕ(x) · ϕ(T x) · . . . · ϕ T
(n)
ϕ (x) = 1,
⎪
#
$−1
⎩
(ϕ(T n x) · . . . · ϕ T −1 x)
,
if n > 0,
if n = 0,
if n < 0.
Thus, we call a cocycle any measurable function ϕ : X → G as well. Given ϕ, consider
a G-extension of T , acting on (X × G, µ ⊗ λG ), defined by the formula
Tϕ (x, g) = (T x, ϕ(x)g).
A cocycle ϕ is called a T -coboundary (or simply a coboundary) if it is of the form
ϕ(x) = ξ(x)(ξ(T x))−1
for some measurable function ξ : X → G (called a transfer function). Two cocycles
φ, ψ : X → G are cohomologous if for some measurable function f : X → G we
have
(58)
( f (T x))−1 φ(x) f (x) = ψ(x).
Analogous notions to the above ones are also present in topological dynamics. Let
T : X → X be a minimal homeomorphism of a compact metric space. Let ϕ : X → G
be a continuous function. We say that ϕ is a topological T -coboundary if it is a
measurable T -coboundary with a continuous transfer function.
14 We use multiplicative notation in G.
123
The Möbius function and continuous extensions of rotations
Compact group extensions: ergodicity and minimality
Let G be a compact Abelian metrizable group and assume that T ∈ Aut(X, B, µ) is
ergodic. Let ϕ : X → G be a cocycle. Then
Tϕ is ergodic if and only if the trivial solution χ = 1, ξ =
+ and
m const is the only solution of χ ◦ ϕ = ξ/ξ ◦ T in χ ∈ G
measurable ξ : X → S1 .
(59)
In particular (see [11]),
If T x = x + α, ϕ : T → S1 is Lipschitz with non-zero degree
then Tϕ is ergodic.
(60)
If Tϕ is ergodic, we will also say that ϕ is ergodic (for T ).
Recall that G acts on X × G via g -→ Rg , where:
.
Rg (x, h) := x, hg −1
for (x, h) ∈ X × G, g ∈ G.
Ergodic components Let ϕ : X → G be a cocycle. Let P(Tϕ , B, µ) stand for the set
of Tϕ -invariant Borel measures whose projection on X is µ. Fix an ergodic measure
λ ∈ P(Tϕ , B, µ). Denote by H be the stabilizer of λ in G, i.e.
H := {g ∈ G : λ ◦ Rg = λ}.
Notice that
7
8
+ : χ ◦ ϕ is a coboundary
0 := χ ∈ G
+ and let
is a (closed) subgroup of G
F = ann 0 := {g ∈ G : for each χ ∈ 0, χ (g) = 1} .
Proposition 2.5.9 (see e.g. [22]) The system (X × G, Tϕ , λ) is isomorphic to (X ×
H, Tψ , µ ⊗ λ H ) for some ergodic ψ : X → H . Moreover, F = H .
Remark 2.5.10 In view of (59), Tϕ is ergodic if and only if 0 = {1}.
Minimal components Let T : X → X be a minimal homeomorphism of a compact
metric space. Let ϕ : X → G be a continuous function and let M be a minimal
component of Tϕ , i.e. M ⊂ X × G is closed and invariant with no proper subsets
having the same properties. Let Htop be the stabilizer of M in G, i.e.
Htop := {g ∈ G : Rg (M) = M}
123
J. Kułaga-Przymus, M. Lemańczyk
(this definition is independent of the initial choice of M). Notice that
+ : χ ◦ ϕ is a topological coboundary}
0top := {χ ∈ G
+ and let
is a closed subgroup of G
Ftop := ann 0top = {g ∈ G : for each χ ∈ 0top , χ (g) = 1}.
Lemma 2.5.11 ([27]) There exists a continuous map τ : X → G/Htop such that
τ (T x) = ϕ(x)τ (x).
(61)
Moreover, M := ∪x∈X {x} × τ (x) is a minimal set.
The proof of the following result is analogous to the one in the measure-theoretical
case. We include it here for the sake of completeness.
Proposition 2.5.12 Htop = Ftop .
Proof Let g0 ∈ Htop and let χ ∈ 0top , i.e.
χ ◦ ϕ(x) = h(T x)/ h(x)
for some continuous function h : X → S1 . We define w : X × G → S1 :
w(x, g) = h(x)−1 χ (g).
This function is continuous and Tϕ -invariant, whence it is constant on each minimal component. It follows that h(x)−1 χ (g)χ (g0 ) = w(x, gg0 ) = w(x, g) =
h(x)−1 χ (g), which implies χ (g0 ) = 1. Therefore g0 ∈ Ftop .
+ such that
Suppose now that there exists g0 ∈ Ftop \Htop . Then there exists χ ∈ G
χ (g0 ) ̸= 1 and χ (Htop ) = {1}. Let τ : X → G/Htop satisfy (61). It follows that χ ◦ τ
is well-defined and we obtain
χ ◦ τ (T x) = χ ◦ ϕ(x) · χ ◦ τ (x)
which implies
χ ◦ ϕ(x) =
χ ◦ τ (T x)
,
χ ◦ τ (x)
i.e. χ ∈ 0top and consequently χ (g0 ) = 1 which is a contradiction.
2
⊓
Relation between the ergodic and the minimal components Let T be a minimal homeomorphism of a compact metric space X and let µ be a T -invariant probability Borel
measure, ergodic with respect to T . Let ϕ : X → G be continuous. Then Tϕ is a homeomorphism of X × G and Tϕ ∈ Aut(X × G, B ⊗B(G), µ⊗λG ). Let λ ∈ P(Tϕ , B, µ).
There are two natural partitions associated to Tϕ :
123
The Möbius function and continuous extensions of rotations
• Perg is the partition into the ergodic components of Tϕ ,
• Pmin is the partition into the minimal components of Tϕ .
Partition Perg is clearly measurable.
Proposition 2.5.13 Partition Pmin is measurable.
Proof Let M ⊂ X ×G be as in Lemma 2.5.11, in particular, M is a minimal component
of Tϕ . Let s : G/Htop → G be a Borel selector of the canonical projection π : G →
G/Htop . Consider η : X → G given by η = s ◦ τ . We obtain
ϕ ′ (x) := ϕ(x)η(x)(η(T x))−1 ∈ Htop
and
M=
D
x∈X
$
#
{x} × η(x)Htop ,
so the map
(x, h) -→ (x, η(x)h)
(62)
settles an equivariant Borel isomorphism of X × H (considered with Tϕ ′ ) and M
(considered with Tϕ ). Moreover, (62) can be naturally extended to a Borel isomorphism
of X × Htop × G/Htop and X × G. Clearly the partition of X × Htop × G/Htop given
by relevant translations of X × Htop × {1} (indexed by G/Htop ) is measurable for
the product measure µ ⊗ λ Htop ⊗ λG/Htop . Hence its image by the Borel extensions
of (62) is also a measurable partition for the image of the measure µ⊗λ Htop ⊗λG/Htop .
This image is equal to µ ⊗ λG , so the partition into minimal components is indeed
measurable.
Remark 2.5.14 Since the partition into the ergodic components can be defined as
the finest measurable partition whose atoms are invariant under the action of the
homeomorphism in question, Perg is finer than Pmin .
As a direct consequence of the above remark we obtain the following:
Remark 2.5.15 We have H ⊃ Htop . The condition H = Htop is necessary and sufficient for the ergodic components of Tϕ to be the same as its minimal components.
Moreover, H = Htop if and only if 0 = 0top .
Unique ergodicity
Proposition 2.5.16 [Furstenberg, see the proof of Theorem I.4 in [12] and Proposition
3.10 in [13]] Let T : X → X be uniquely ergodic and let ϕ : X → G be a continuous
cocycle with values in a compact Abelian group. If Tϕ is ergodic with respect to µ⊗λG
then it is uniquely ergodic.
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J. Kułaga-Przymus, M. Lemańczyk
A remark on the KBSZ criterion for T x = x + α
Let T x = x + α be an irrational rotation on T.
Proposition 2.5.17 Let A ⊂ Z.
1. If A is such that
r A ∩ s A = ∅ for sufficiently large prime numbers r ̸= s,
(63)
then (7) holds true for every f ∈ C(T) such that supp +
f := {n ∈ Z : +
f (n) ̸=
0} = A.
2. Suppose that A contains infinitely many primes. Let f ∈ C(T) be such that
supp +
f = A and all nonzero Fourier coefficients are positive. Then (7) fails for f .
Remark 2.5.18 Note that every finite set satisfies (63), so all trigonometric polynomials satisfy (7); the set {2n : n ≥ 1} is an example of an infinite set satisfying (63).
Proof of Proposition 2.5.17 We will consider the behavior of the sums in (7) at (0, 0).
Given two different prime numbers r, s, we set Ir,s := {(x, y) : sx = r y}, which is a
closed subgroup of T2 (of course (0, 0) ∈ Ir,s ), invariant under T r × T s . It is not hard
to see that
1 "
δr nα,snα → λ Ir,s
(64)
N
n≤N
and that
W : Ir,s → T, W (x, y) = ax + by (ar + bs = 1)
is a continuous group isomorphism.
(65)
In particular, W sends λ Ir,s to λT . In view of (64),
%
1 "
rn
sn
f (T 0) f (T 0) →
f ⊗ f dλ Ir,s .
N
Ir,s
n≤N
Since W −1 (t) = (r t, st), it follows that
%
Ir,s
f ⊗ f dλ Ir,s =
Now, if f (t) =
!
n∈Z cn e
2πint
f (r t) =
and we can compute
123
*
Ir,s
%
"
n∈Z
T
( f ⊗ f ) ◦ W −1 (t) dt =
%
T
f (r t) f (st) dt.
(66)
then
cn ne2πir nt ,
f (st) =
"
cn e2πisnt
n∈Z
f ⊗ f dλ Ir,s using (66), (67) and Parseval’s formula.
(67)
The Möbius function and continuous extensions of rotations
1. Fix infinitely many pairs (r j , s j ) ∈ A × A of
* distinct prime numbers, r j , s j → ∞.
Take j ≥ 1. It is enough to show that Ir ,s f ⊗ f dλ Ir j ,s j ̸= 0. By Parsej
j
*val’s formula, and the fact that all Fourier coefficients of f are positive, we have
Ir ,s f ⊗ f dλ Ir j ,s j ≥ 0. Since r j s j = s j r j , the s j r j th Fourier coefficient of
j
j
f (r j ·) is cs j > 0, while the r j s j th Fourier coefficient of f (s j ·) is cr j > 0. Hence,
*
by Parseval’s formula, Ir ,s f ⊗ f dλ Ir j ,s j > 0.
j
j
2. Each (x, y) ∈ T2 belongs to a coset of Ir,s . The proof for an arbitrary coset of
Ir,s goes along the same lines as for Ir,s itself (on the cosets of Ir,s , we consider
translations of Haar measure λ Ir,s and W is practically the same). Since r A ∩ s A =
*∅, the supports of the Fourier transforms of f (r ·) and f (s·) are disjoint, whence
2
⊓
T f (r t) f (st) dt = 0.
References
1. El Abdalaoui, E.H., Kasjan, S., Lemańczyk, M.: 0-1 Sequences of the Thue-Morse type and Sarnak’s
conjecture. arXiv:1304.3587 (To appear in Proc. Amer. Math. Soc)
2. El Abdalaoui, E.H., Lemańczyk, M., de la Rue, T.: On spectral disjointness of powers for rank-one
transformations and Möbius orthogonality. J. Funct. Anal. 266, 284–317 (2014)
3. Anzai, H.: Ergodic skew product transformations on the torus. Osaka Math. J. 3, 83–99 (1951)
4. Auslander, J.: On the proximal relation in topological dynamics. Proc. Amer. Math. Soc. 11, 890–895
(1960)
5. Bourgain, J.: Möbius-Walsh correlation bounds and an estimate of Mauduit and Rivat. J. Anal. Math.
119, 147–163 (2013)
6. Bourgain, J.: On the correlation of the Moebius function with rank-one systems. J. Anal. Math. 120,
105–130 (2013)
7. Bourgain, J., Sarnak, P., Ziegler, T.: Disjointness of Möbius from horocycle flows. In: From Fourier
analysis and number theory to radon transforms and geometry, Dev. Math., vol. 28, pp. 67–83. Springer,
New York (2013)
8. Dartyge, C., Tenenbaum, G.: Sommes des chiffres de multiples d’entiers. Ann. Inst. Fourier (Grenoble)
55, 2423–2474 (2005)
9. Davenport, H.: On some infinite series involving arithmetical functions. II. Quart. J. Math. Oxford, 8,
313–320 (1937)
10. Ellis, R.: A semigroup associated with a transformation group. Trans. Amer. Math. Soc. 94, 272–281
(1960)
11. Furstenberg, H.: Strict ergodicity and transformation of the torus. Am. J. Math. 83, 573–601 (1961)
12. Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)
13. Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. Princeton University
Press, Princeton (1981) (M. B. Porter Lectures)
14. Green, B.: On (not) computing the Möbius function using bounded depth circuits. Combin. Probab.
Comput. 21, 942–951 (2012)
15. Green, B., Tao, T.: The Möbius function is strongly orthogonal to nilsequences. Ann. of Math. 175(2),
541–566 (2012)
16. Indlekofer, K.-H., Kátai, I.: Investigations in the theory of q-additive and q-multiplicative functions.
I. Acta Math. Hungar. 91, 53–78 (2001)
17. Kátai, I.: A remark on a theorem of H. Daboussi. Acta Math. Hungar. 47, 223–225 (1986)
18. Katok, A.: Combinatorial constructions in ergodic theory and dynamics. University Lecture Series,
vol. 30. American Mathematical Society, Providence, RI (2003)
19. Keane, M.: Generalized Morse sequences. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10, 335–
353 (1968)
20. Khinchin, A.Y.: Continued fractions. Dover Publications Inc., Mineola (1997) (Reprint of the 1964
translation)
123
J. Kułaga-Przymus, M. Lemańczyk
21. Kwiatkowski, J., Lemańczyk, M., Rudolph, D.: A class of real cocycles having an analytic coboundary
modification. Israel J. Math. 87, 337–360 (1994)
22. Lemańczyk, M., Mentzen, M.K.: Compact subgroups in the centralizer of natural factors of an ergodic
group extension of a rotation determine all factors. Ergodic Theory Dynam. Syst. 10, 763–776 (1990)
23. Liu, J., Sarnak, P.: The Möbius function and distal flows. arXiv:1303.4957 (2013)
24. Mauduit, C., Rivat, J.: Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Ann.
Math. 171(2), 1591–1646 (2010)
25. Mauduit, C., Rivat, J.: Prime numbers along Rudin-Shapiro sequences. J. Eur. Math. Soc. (to appear)
26. Sarnak, P.: Mobius randomness and dynamics. Not. S. Afr. Math. Soc. 43, 89–97 (2012)
27. Siemaszko, A.: Ergodic and topological properties of distal systems. Ph.D. thesis (1996)
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