Digit frequencies and self-affine sets with non-empty interior

0.66 MB·

by  Simon Baker

DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY
INTERIOR
SIMON BAKER
7 Abstract. In this paper we study digit frequencies in the setting of expansions in non-
1 integer bases, and self-affine sets with non-empty interior.
0
Within expansions in non-integer bases we show that if β ∈ (1,1.787...) then every
2
√
x ∈ (0, 1 ) has a simply normal β-expansion. We also prove that if β ∈ (1,1+ 5) then
n β−1 2
a every x ∈ (0, 1 ) has a β-expansion for which the digit frequency does not exist, and
β−1
J
a β-expansion with limiting frequency of zeros p, where p is any real number sufficiently
4
close to 1/2.
2
For a class of planar self-affine sets we show that if the horizontal contraction lies in

]
S a certain parameter space and the vertical contractions are sufficiently close to 1, then
D every nontrivial vertical fibre contains an interval. Our approach lends itself to explicit
h. calculation and give rise to new examples of self-affine sets with non-empty interior. One
t particular strength of our approach is that it allows for different rates of contraction in
a
m the vertical direction.
[

1
v
3 1. Introduction
7
7 Let x ∈ [0,1]. A sequence ((cid:15) ) ∈ {0,1}N is called a binary expansion of x if
6 i
0
∞
1. x = (cid:88) (cid:15)i.
0 2i
7 i=1
1 It is well known that apart from the dyadic rationals (numbers of the form p/2n) every
:
v
x ∈ [0,1] has a unique binary expansion. The exceptional dyadic rationals have precisely
i
X
two binary expansions. A seemingly innocuous generalisation of these representations is
r
a to replace the base 2 with a parameter β ∈ (1,2). That is, given x ∈ R, we call a sequence
((cid:15) ) ∈ {0,1}N a β-expansion of x if
i
∞
(cid:88) (cid:15)
i
x = π (((cid:15) )) := .
β i βi
i=1
Date: 25th January 2017.
2010 Mathematics Subject Classification. Primary 11A63; Secondary 28A80, 11K55.
Key words and phrases. Expansions in non-integer bases, Digit frequencies, Self-affine sets.
1
2 SIMON BAKER
These representations were first introduced in the papers of Parry [28] and R´enyi [29]. It
is straightforward to show that x has a β-expansion if and only if x ∈ [0, 1 ]. In what
β−1
follows we will let I := [0, 1 ].
β β−1
Despite being a simple generalisation of binary expansions, β-expansions exhibit far
more exotic behaviour. In particular, one feature of β-expansions that makes them an
interesting object to study, is that an x ∈ I may have many β-expansions. In fact a result
β
of Sidorov [30] states that for any β ∈ (1,2), Lebesgue almost every x ∈ I has a continuum
β
of β-expansions. Moreover, for any k ∈ N∪{ℵ }, there exists β ∈ (1,2) and x ∈ I such
0 β
that x has precisely k β-expansions, see [7, 9, 15, 16, 31]. Note that the endpoints of I
β
always have a unique β-expansion for any β ∈ (1,2).
A particularly useful technique for studying both binary expansions and β-expansions is
to associate a dynamical system to the base. One can then often reinterpret a problem in
terms of a property of the dynamical system. The underlying geometry of the dynamical
system can then make a problem much more tractable. In this paper we prove results
relating to digit frequencies and self-affine sets. These results are of independent interest
but also demonstrate the strength of the dynamical approach to β-expansions.
2. Statement of results
2.1. Digit frequencies. Let ((cid:15) ) ∈ {0,1}N. We define the frequency of zeros of ((cid:15) ) to be
i i
the limit
#{1 ≤ i ≤ n : (cid:15) = 0}
i
freq ((cid:15) ) := lim .
0 i n→∞ n
Assumingthelimitexists. Wecallasequence((cid:15) )simply normal iffreq ((cid:15) ) = 1/2.Foreach
i 0 i
x ∈ [0,1], we let freq (x) denote the frequency of zeros in its binary expansion whenever
0
thelimitexists. Whenthelimitdoesnotexistwesayfreq (x)doesnotexist. Thefollowing
0
results are well known:
(1) Lebesgue almost every x ∈ [0,1] has a simply normal binary expansion.
(2) dim ({x : freq (x) does not exist}) = 1.
H 0
(3) For each p ∈ [0,1] we have
−plogp−(1−p)log(1−p)
dim ({x : freq (x) = p}) = .
H 0 log2
In (3) we have adopted the convention 0log0 = 0. The first statement is a consequence of
Borel’s normal number theorem [10], the second statement appears to be folklore, and the
third statement is a result of Eggleston [14].
DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 3
The above results provide part of the motivation for the present work. In particular,
we are interested in whether analogues of these results hold for expansions in non-integer
bases. Our first result in the setting of β-expansions is the following.
Theorem 2.1. (1) Let β ∈ (1,β ). Then every x ∈ (0, 1 ) has a simply normal
KL β−1
β-expansion.
√
(2) Let β ∈ (1, 1+ 5). Then every x ∈ (0, 1 ) has a β-expansion for which the fre-
2 β−1
quency of zeros does not exist.
√
(3) Let β ∈ (1, 1+ 5). Then there exists c = c(β) > 0 such that for every x ∈ (0, 1 )
2 β−1
and p ∈ [1/2−c,1/2+c], there exists a β-expansion of x with frequency of zeros
equal to p.
The quantity β ≈ 1.787 appearing in statement 1 of Theorem 2.1 is the Komornik-
KL
Loreti constant introduced in [27]. In [27] Komornik and Loreti proved that β is the
KL
smallest base for which 1 has a unique β-expansion. It has since been shown to be im-
portant for many other reasons. We elaborate on the significance of this constant and
its relationship with the Thue-Morse sequence in Section 3. Note that we can explicitly
calculate a lower bound for the quantity c appearing in statement 3 of Theorem 2.1. We
include some explicit calculations in Section 6.
It follows from the results listed above that the set of x whose binary expansion is not
simply normal has Hausdorff dimension 1. Our next result shows that as β approaches 2
we see a similar phenomenon.
Theorem 2.2.
(cid:16)(cid:110) (cid:111)(cid:17)
limdim x : x has no simply normal β-expansion = 1.
H
β(cid:37)2
2.2. Hybrid expansions. In this section we consider β-expansions where our digit set
is {−1,1} instead of {0,1}. Given β ∈ (1,2) and x ∈ [ −1 , 1 ], we say that a sequence
β−1 β−1
((cid:15) ) ∈ {−1,1}N is a hybrid expansion of x if the following holds:
i
∞
(cid:88) (cid:15)
i
x =
βi
i=1
and
n
1 (cid:88)
x = lim (cid:15) .
i
n→∞ n
i=1
Hybrid expansions were first introduced by Gu¨ntu¨rk in [21]. Interestingly, the original
motivation for studying hybrid expansions was to overcome the problem of analogue to
4 SIMON BAKER
digital conversion where the underling system has background noise. In [21] the following
result was asserted without proof.
Theorem 2.3. There exists C > 0, such that for all β ∈ (1,1 + C ) there exists c =
1 1
c(β) > 0, such that every x ∈ [−c,c] has a hybrid expansion.
A proof was subsequently provided by Dajani, Jiang, and Kempton in [12]. They showed
that one can take C ≈ 0.327. We improve upon this theorem in the following way.
1
√
Theorem 2.4. Let β ∈ (1, 1+ 5). Then there exists c = c(β) > 0 such that every x ∈
2
[−c,c] has a hybrid expansion.
It would be desirable to obtain a result of the form: there exists C > 0 such that for
every β ∈ (1,1 + C) every x ∈ (− 1 , 1 ) has a hybrid expansion. However, it is an
β−1 β−1
immediate consequence of the definition that if x has a hybrid expansion then x ∈ [−1,1].
Since [−1,1] (cid:40) ( −1 , 1 ) for all β ∈ (1,2) it is clear that such a result is not possible.
β−1 β−1
Note that if we normalised by a function that decayed at a slower rate than n−1 we would
not necessarily have this obstruction. The following result shows that if we replace n−1
with another normalising function that satisfies a certain growth condition, then we have
our desired result.
√
Theorem 2.5. Let β ∈ (1, 1+ 5). Then there exists c = c(β) > 0 such that if f : N →
2
(0,∞) is a strictly increasing function which satisfies
limsupf(n+1)−f(n) < c
n→∞
and
lim f(n) = ∞,
n→∞
then for every x ∈ (− 1 , 1 ) there exists ((cid:15) ) ∈ {−1,1}N such that
β−1 β−1 i
∞
(cid:88) (cid:15)
i
x =
βi
i=1
and
n
1 (cid:88)
x = lim (cid:15) .
i
n→∞ f(n)
i=1
The following corollary is an immediate consequence of Theorem 2.5.
DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 5
√
Corollary 2.6. Let β ∈ (1, 1+ 5). Then for every x ∈ (− 1 , 1 ) there exists ((cid:15) ) ∈
2 β−1 β−1 i
{−1,1}N such that
∞
(cid:88) (cid:15)
i
x =
βi
i=1
and
n
1 (cid:88)
x = lim (cid:15) .
n→∞ n1/2 i
i=1
2.3. A family of overlapping self-affine sets and simultaneous expansions. Let
{S }m be a collection of contracting maps acting on Rd. A result of Hutchinson [23]
j j=1
states that there exists a unique non-empty compact set Λ ⊆ Rd such that
m
(cid:91)
Λ = S (Λ).
j
j=1
We call Λ the attractor associated to {S }. Often one is interested in determining the
j
topological properties of Λ. When the collection {S } consists solely of similarities than
j
the attractor Λ is reasonably well understood. However, when the collection {S } contains
j
affine maps the situation is known to be much more complicated.
In this paper we focus on the following family of self-affine sets. Let 1 < β ,β ,β ≤ 2
1 2 3
and
(cid:16)x−1 x−1(cid:17) (cid:16)x+1 x+1(cid:17)
S (x,y) = , and S (x,y) = , .
−1 1
β β β β
1 2 1 3
For this collection of contractions we denote the associated attractor by Λ . In Figure
β1,β2,β3
1 we include some examples.
Figure 1. A plot of Λ ,Λ ,Λ
2,1.81,1.66 1.66,1.33,1.53 1.2,1.11,1.05
When β = β we denote Λ by Λ . The case where β = β was studied in
2 3 β1,β2,β3 β1,β2 2 3
[12] and [22]. One problem the authors of these papers were particularly interested in was
6 SIMON BAKER
determining those pairs (β ,β ) for which the attractor Λ has non-empty interior. The
1 2 β1,β2
best result in this direction is the following result due to Hare and Sidorov [22].
Theorem 2.7. If β (cid:54)= β and
1 2
(cid:12)β8 −β8(cid:12) (cid:12)β7β7(β −β )(cid:12)
(2.1) (cid:12) 2 1(cid:12)+(cid:12) 2 1 2 1 (cid:12) ≤ 2.
(cid:12)β7 −β7(cid:12) (cid:12) β7 −β7 (cid:12)
2 1 2 1
Then Λ has non-empty interior and (0,0) ∈ Λo.
β1,β2
Let π denote the projection from R2 onto the x-axis. For each x ∈ π(Λ ) let
β1,β2,β3
Λx := {y ∈ R : (x,y) ∈ Λ }.
β1,β2,β3 β1,β2,β3
We call Λx the fibre of x. Note that π(Λ ) = [ −1 , 1 ]. The following state-
β1,β2,β3 β1,β2,β3 β1−1 β1−1
ment is our main result for Λ .
β1,β2,β3
√
Theorem 2.8. Let β ∈ (1, 1+ 5). Then there exists c = c(β ) > 0 such that for all
1 2 1
β ,β ∈ (1,1+c) and x ∈ (− 1 , 1 ) the fibre Λx contains an interval. Moreover
2 3 β1−1 β1−1 β1,β2,β3
Λ has non-empty interior and (0,0) ∈ Λo
β1,β2,β3 β1,β2,β3
We emphasise that Theorem 2.8 covers the case where β (cid:54)= β . Our approach lends
2 3
itself to explicit calculation and following our method one can obtain a lower bound for
the value c appearing in Theorem 2.8. We include some explicit calculations in Section 6.
√
Note that for any β sufficiently close to 1+ 5 the set of β ∈ (1,2) satisfying (2.1)
1 2 2
is empty. Consequently Theorem 2.8 provides new examples of β ,β for which Λo is
1 2 √ β1,β2
non-empty. Theorem 2.8 is also optimal in the following sense. For any β ∈ [1+ 5,2) and
1 2
β ,β ∈ (1,2), there exists x ∈ (− 1 , 1 ) such that the fibre Λx is countable and
2 3 β1−1 β1−1 β1,β2,β3
therefore does not contain an interval. We explain why this is the case in Section 7.
It is natural to ask whether the property Λx contains an interval for every x ∈
β1,β2,β3
(− 1 , 1 ) is stronger than the property Λo (cid:54)= ∅. This is in fact the case and is a
β1−1 β1−1 β1,β2,β3
consequence of the following proposition.
Proposition 2.9. Λo (cid:54)= ∅ if and only if {x : Λx contains an interval} contains
β1,β2,β3 β1,β2,β3
an open dense subset of [ −1 , 1 ].
β1−1 β1−1
Proof. Letusstartbyintroducingsomenotation. LetF = {x : Λx contains an interval}.
β1,β2,β3
Suppose Λo (cid:54)= ∅. Then there exists I and J two nontrivial open intervals such that
β1,β2,β3
I×J ⊆ Λ . Let φ (x) = x−1 and φ (x) = x+1. Since S (I×J) is an open rectangle
β1,β2,β3 −1 β1 1 β1 −1
contained in Λ , it follows that φ (I) ⊆ F. Similarly φ (I) ⊆ F. Repeating this
β1,β2,β3 −1 1
argument, it follows that all images of I under finite concatenations of φ and φ are
−1 1
DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 7
contained in F. The union of these images of I is an open dense subset of [ −1 , 1 ]. It
β1−1 β1−1
follows that F contains an open dense subset of [ −1 , 1 ].
β1−1 β1−1
It remains to prove the leftwards implication. We start by partitioning the set F. Given
(a,b,c,d) ∈ Z4 let
(cid:110) (cid:104)a c(cid:105) (cid:111)
F := x : , ⊆ Λx .
a,b,c,d b d b1,b2,b3
Importantly we have
(cid:91)
F = F .
a,b,c,d
(a,b,c,d)∈Z4,a/b<c/d
Suppose F is nowhere dense for all (a,b,c,d) ∈ Z4. Since F contains an open dense set
a,b,c,d
its complement is a nowhere dense set. It follows that [ −1 , 1 ] is the countable union
β1−1 β1−1
of nowhere dense sets. By the Baire category theorem this is not possible. Therefore there
must exist (a(cid:48),b(cid:48),c(cid:48),d(cid:48)) ∈ Z4 such that a(cid:48)/b(cid:48) < c(cid:48)/d(cid:48) and F is dense in some non
a(cid:48),b(cid:48),c(cid:48),d(cid:48)
trivial interval I(cid:48). Since Λ is closed it follows that
β1,β2,β3
I(cid:48) ×[a(cid:48)/b(cid:48),c(cid:48)/d(cid:48)] ⊆ Λ
β1,β2,β3
and Λ has non-empty interior. (cid:3)
β1,β2,β3
Interestingly computer simulations suggest that there exist examples where Λo (cid:54)= ∅
β1,β2,β3
yet {x : Λx is a singleton} is infinite and even has positive Hausdorff dimension. See
β1,β2,β3
Figure 2 for such an example.
Figure 2. A plot of Λ . For this choice of β and β it can be shown
1.8,1.05 1 2
that {x : Λx is a singleton} has positive Hausdorff dimension.
β1,β2
8 SIMON BAKER
In [21], in addition to the notion of a hybrid expansion, Gu¨ntu¨rk introduced the notion
of a simultaneous expansion. These are defined as follows. Given x ∈ [ −1 , 1 ] and
β1−1 β1−1
β ,β ∈ (1,2), we say that a sequence ((cid:15) ) ∈ {−1,1}N is a simultaneous (β ,β ) expansion
1 2 i 1 2
of x if
∞ ∞
(cid:88) (cid:15) (cid:88) (cid:15)
i i
x = = .
βi βi
i=1 1 i=1 2
These expansions relate to our self-affine set via the following observation. If β = β then
2 3
∞ ∞
(cid:110)(cid:16)(cid:88) (cid:15) (cid:88) (cid:15) (cid:17) (cid:111)
i i N
Λ = , : ((cid:15) ) ∈ {−1,1} .
β1,β2 βi βi i
i=1 1 i=1 2
Therefore
(cid:110) (cid:111)
(x,x) : x has a simultaneous (β ,β ) expansion = Λ ∩{(x,x) : x ∈ R}.
1 2 β1,β2
In [21] it was asserted by Gu¨ntu¨rk that there exists C > 0, such that for 1 < β < β <
1 2
1 + C, there exists c = c(β ,β ) > 0 such that every x ∈ (−c,c) has a simultaneous
1 2
(β ,β ) expansion. Note that the existence of C > 0 satisfying the above follows if one
1 2
can show that for 1 < β < β < 1+C the attractor Λ contains (0,0) in its interior.
1 2 β1,β2
Using this observation Gu¨ntu¨rk’s assertion was proved to be correct in [12]. The largest
parameter space for which it is known that (0,0) ∈ Λo , and consequently that any x
β1,β2
sufficiently close to zero has a simultaneous (β ,β ) expansion, is that stated in Theorem
1 2
2.7. Ourcontributioninthisdirectionisthefollowingtheoremthatfollowsasanimmediate
consequence of Theorem 2.8 by taking β = β .
2 3
√
Theorem 2.10. Let β ∈ (1, 1+ 5). Then there exists C = C(β ) > 0 such that if
1 2 1
β ∈ (1,1+C), then every x sufficiently small has a simultaneous (β ,β )-expansion.
2 1 2
Before moving onto our proofs we say a few words about the methods used in this paper
and compare them with those used in [12] and [22]. In these papers the authors show that
(0,0) ∈ Λo by constructing a polynomial P(x) = xn+b xn−1+b x+b which satisfies:
β1,β2 n−1 1 0
(1) P(β ) = P(β ) = 0
1 2
(2) (cid:80)n−1|b | ≤ 2
j=0 j
(3) b = 0
1
(4) b (cid:54)= 0.
0
Once the existence of this polynomial is established, one can devise an algorithm which
can be applied to any x ,x sufficiently small, this algorithm then yields an ((cid:15) ) ∈ {−1,1}N
1 2 i
such that (x ,x ) = ((cid:80)∞ (cid:15) β−i,(cid:80)∞ (cid:15) β−i).
1 2 i=1 i 1 i=1 i 2
DIGIT FREQUENCIES AND SELF-AFFINE SETS WITH NON-EMPTY INTERIOR 9
This approach is somewhat unsatisfactory. The existence of the polynomial and the
algorithm used to construct the ((cid:15) ) provide little intuition as to why (0,0) should be in
i
the interior of Λ . Our approach, as well as allowing for different rates of contraction
β1,β2
in the vertical direction, is more intuitive and explicitly constructs the interval appearing
in each fibre of Λ .
β1,β2,β3
The rest of this paper is arranged as follows. In Section 3 we recall and prove some
technical results that are required to prove our theorems. In Section 4 we prove our
theorems relating to digit frequencies. In Section 5 we prove Theorem 2.8. In Section 6 we
include an example where we explicitly calculate some of the parameters appearing in our
theorems. In Section 7 we include some general discussion and pose some questions.
3. Preliminaries
In this section we prove some useful technical results and recall some background mater-
ial. Let us start by introducing the maps T (x) = βx+1, T (x) = βx and T (x) = βx−1.
−1 0 1
Given an x ∈ I we let
β
∞
(cid:110) (cid:88) (cid:15) (cid:111)
N i
Σ (x) := ((cid:15) ) ∈ {0,1} : = x
β i βi
i=1
and
(cid:110) (cid:111)
Ω (x) := (a ) ∈ {T ,T }N : (a ◦···◦a )(x) ∈ I for all n ∈ N .
β i 0 1 n 1 β
Similarly, given x ∈ I(cid:101) := [ −1 , 1 ] let
β β−1 β−1
∞
(cid:110) (cid:88) (cid:15) (cid:111)
N i
Σ(cid:101) (x) := ((cid:15) ) ∈ {−1,1} : = x
β i βi
i=1
and
(cid:110) (cid:111)
Ω(cid:101) (x) := (a ) ∈ {T ,T }N : (a ◦···◦a )(x) ∈ I(cid:101) for all n ∈ N .
β i −1 1 n 1 β
The dynamical interpretation of β-expansions is best seen through the following result.
Lemma 3.1. For any x ∈ I (x ∈ I(cid:101) ) we have Card Σ (x) = Card Ω (x)(Card Σ(cid:101) (x) =
β β β β β
Card Ω(cid:101) (x)). Moreover, the map which sends ((cid:15) ) to (T ) is a bijection between Σ (x) and
β i (cid:15)i β
Ω (x)(Σ(cid:101) (x) and Ω(cid:101) (x)).
β β β
Lemma 3.1 was originally proved in [6] for an arbitrary digit set of the form {0,...,m}.
The proof easily extends to the digit set {−1,1}.
Lemma 3.1 allows us to reinterpret problems from β-expansions in terms of the allowable
trajectories that can occur within a dynamical system. In Figure 3 we include a graph of T
0
and T acting on I . One can see from this picture, or check by hand, that if x ∈ [1, 1 ]
1 β β β(β−1)
10 SIMON BAKER
0 1
1 1 β−1
β β(β−1)
Figure 3. The overlapping graphs of T and T .
0 1
then both T and T map x into I . Therefore, by Lemma 3.1, this x has at least two
0 1 β
β-expansions. More generally, if there exists a sequence of T ’s and T ’s that map x into
0 1
[1, 1 ], then x has at least two β-expansions.
β β(β−1)
The interval [1, 1 ] is clearly important when it comes to studying Σ (x) and Ω (x).
β β(β−1) β β
In what follows we let
(cid:104)1 1 (cid:105)
S := , .
β
β β(β −1)
Another particularly useful interval for studying β-expansions is
(cid:104) 1 β (cid:105)
O := , .
β β2 −1 β2 −1
The analogues of S and O for the digit set {−1,1} are
β β
(cid:104) β −2 2−β (cid:105) (cid:104) 1−β β −1 (cid:105)
S(cid:101) := , and O(cid:101) := , .
β β(β −1) β(β −1) β β2 −1 β2 −1