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Department for Analysis and Computational Number Theory
Non-normal numbers
The interplay of symbolic and topological dynamical systems
Manfred G. Madritsch
Department for Analysis and Computational Number Theory
Graz University of Technology
[email protected]
Combinatorics, Automata and Number Theory 2012
Supported by the Austrian Science Fund (FWF), project S9603.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
1 / 29
Department for Analysis and Computational Number Theory
Outline
Introduction
The non-normal numbers
Normal numbers and fibered systems
The Cantor numeration system
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
2 / 29
Department for Analysis and Computational Number Theory
The q-adic representation
Every real x ∈ [0, 1] can be written in the form
X
x=
xh q −h = 0.x1 x2 x3 . . .
h≥1
with digits xh ∈ Dq := {0, 1, . . . , q − 1} for h ≥ 1.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
3 / 29
Department for Analysis and Computational Number Theory
Normal numbers
Definition (Wall 1949)
We say that a real x ∈ [0, 1] is normal with respect to the base q if
the sequence q n x (mod 1) is uniformly distributed in [0, 1], i.e.
#{n ≤ N : a ≤ {q n x} < b}
=b−a
N→∞
N
lim
for all 0 ≤ a < b ≤ 1.
Definition (Niven-Zuckerman 1951)
We say that a real x ∈ [0, 1] is normal with respect
the base q if in
P∞ xto
n
its q-adic representation x = 0.x1 x2 x3 . . . = n=1 qn (xn ∈ Z,
0 ≤ xn ≤ q − 1), every block of k digits appears with the asymptotic
frequency q1k (k = 1, 2, . . .).
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
4 / 29
Department for Analysis and Computational Number Theory
Normal numbers
The most well known example of a construction of a number number
is due to Champernowne (1933)
0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 . . .
There are generalizations of this construction with
I
prime numbers (Copeland et Erdős; Nakai et Shiokawa),
I
polynomial values (Davenport et Erdős; Naka et Shiokawa;
Madritsch, Thuswaldner et Tichy),
and generalizations to different number systems like
I
matrix number systems (Madritsch),
I
number systems in algebraic fields (Madritsch),
I
β number systems (Ito et Shiokawa).
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
5 / 29
Department for Analysis and Computational Number Theory
Outline
Introduction
The non-normal numbers
Normal numbers and fibered systems
The Cantor numeration system
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
6 / 29
Department for Analysis and Computational Number Theory
The difference between
the normal and non-normal numbers
Theorem (Oxtoby et Ulam 1941)
The set of normal numbers is of first category in R/Z.
The set of non-normal numbers is
I small from the measure theoretic point of view and
I big from the topologic point of view.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
7 / 29
Department for Analysis and Computational Number Theory
The frequency of digits
Suppose that x ∈ [0, 1] and b is a sequence of k digits in base q.
We denote by Π(x, b, n) the frequency of the block
b = b1 b2 . . . bk among the first n digits of x, i.e.
Π(x, b, n) =
#{0 ≤ i < n : xi+1 = b1 , . . . , xi+k = bk }
.
n
Furthermore let Πk (x, n) denote the vector of frequencies
Π(x, b, n) of all blocks b of length k, i.e.
Πk (x, n) = (Π(x, b, n))b∈Dk .
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
8 / 29
Department for Analysis and Computational Number Theory
Essentially non-normal numbers
Definition
We call a number x essentially non-normal in base q if for every
digit i ∈ D the limiting frequency
lim Π(x, i, k)
k→∞
does not exist.
Theorem (Albeverio, Pratsiovytyi and Torbin, 2005)
The set of essentially non-normal numbers
I is of second category and
I has Hausdorff dimension 1.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
9 / 29
Department for Analysis and Computational Number Theory
Extremely non-normal numbers
Definition
We call x extremely non-normal in base q if every shift invariant
probability vector is a limit point of (Πk (x, n))n .
Theorem (Olsen, 2004)
The set of extremely non-normal numbers
I is of second category and
I has Hausdorff dimension 0.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
10 / 29
Department for Analysis and Computational Number Theory
Outline
Introduction
The non-normal numbers
Normal numbers and fibered systems
The Cantor numeration system
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
11 / 29
Department for Analysis and Computational Number Theory
Fibered systems
Let B be a set and T : B → B be a transformation. We call the
pair (B, T ) a fibered system if the following conditions are
satisfied:
1. There exists an at most countable set D, which we call the
set of digits.
2. There is an application k : B → D such that the sets
B(i) = k −1 ({i}) = {x ∈ B : k(x) = i}
form a partition of B.
3. The restriction of T to B(i) is injective for every i ∈ D.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
12 / 29
Department for Analysis and Computational Number Theory
Fibered systems
In all our examples let B = [0, 1].
1. Let q ≥ 2 be an integer. The sets B(i) = [ qi , i+1
] for
q
i ∈ {0, 1, . . . , q − 1} together with the transformation
T (x) = qx − bqxc provide us the q-adic representation of x.
1
2. The sets B(i) = [ i+1
, 1i ] for i ∈ N together with the
transformation T (x) = x1 − b x1 c provide the continued
fraction representation.
1
3. The sets B(i) = [ i+1
, 1i ] together with the transformation
T (x) = b x1 c b x1 c + 1 x − b x1 c provide the Lüroth
representation.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
13 / 29
Department for Analysis and Computational Number Theory
The representation is an infinite word
One can see the q-adic representation of x as an infinite word
over the alphabet {0, . . . , q − 1}.
In particular, the q-adic representation x = 0.x1 x2 x3 . . . assigns
the infinite word ω(x) = ω = x1 x2 x3 . . . to x.
By the definition of Niven and Zuckermann we call a number x
normal if for each word b of length k over the alphabet
D := {0, 1, . . . , q − 1} the frequency of occurrences of this word
as a factor of ω tends to q −k .
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
14 / 29
Department for Analysis and Computational Number Theory
The frequency of a factor
Suppose that ω = ω1 ω2 ω3 . . . is an infinite word over the
alphabet D. We write P(ω, b, n) for the frequency of the factor
b = b1 b2 . . . bk among the first n letters of ω, i.e.
P(ω, b, n) =
#{0 ≤ i < n : ωi+1 = b1 , . . . , ωi+k = bk }
.
n
Furthermore let Pk (ω, n) denote the vector of frequencies
P(ω, b, n) for all factors b of length k, i.e.
Pk (ω, n) = (P(ω, b, n))b∈Dk .
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
15 / 29
Department for Analysis and Computational Number Theory
The specification property
Definition
A transformation T : X → X satisfies the specification property
if for every ε > 0 there exists an integer M > 0 such that: for
any pair of points x1 , x2 ∈ X , for any positive integers
a1 < b1 < a2 < b2 such that a2 − b1 > M, and for any integer
p > b2 − a1 + M, there exists x ∈ X such that T p x = x and
such that
d(T j x, T j x1 ) < ε
j
j
and d(T x, T x2 ) < ε
M.G. Madritsch
Non-normal numbers
for a1 ≤ j ≤ b1
for a2 ≤ j ≤ b2 .
CANT, 21 may 2012
16 / 29
Department for Analysis and Computational Number Theory
The specification property
Theorem (Sigmund 1974)
If T satisfies the specification property,
then the set of normal numbers is of first category.
This property is satisfied for:
I the transformation of the q-adic representation,
I the shift over a finite or infinite alphabet and many
subshifts,
I the hyperbolic automorphisms of Tn (i.e. those which are
defined by the elements of SL(n, Z) which have no
eigenvalue on the unit circle.
etc.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
17 / 29
Department for Analysis and Computational Number Theory
The symbolic construction
In order to prove that the set of essentially non-normal numbers
is of second category, one constructs and investigates the sets Cn
defined as
Cn = x ∈ [0, 1] :
. . 1} . . . (q − 1) . . . (q − 1) β1 β2 β3 . . .
x = 0.α1 . . . αn |0 .{z
. . 0} |1 .{z
|
{z
}
n
M.G. Madritsch
n
Non-normal numbers
n
CANT, 21 may 2012
18 / 29
Department for Analysis and Computational Number Theory
The symbolic construction
On the other side, in order to prove this result for the set of
extremely non-normal numbers, we start with the definition of
two useful sets:
Zn = Zn (c, p, k) =
cn = Z
cn (p, k) =
Z






M.G. Madritsch
ω
ω
`
∈
D kPk (ω) − pk ≤
`≥knq k
[
`
∈
D kPk (ω) − pk ≤
`≥knq k
[
Non-normal numbers

1
,
n

6
.
n
CANT, 21 may 2012
19 / 29
Department for Analysis and Computational Number Theory
The symbolic construction
The central idea consists in the following lemma.
Lemma
For every positive integer n there exist functions un : D∗ → D∗
and Qn : D∗ → N such that
π([un (ω)]) ⊂ π([ω])◦ ,
cn ,
un (ω) Zn · · · Zn ⊂ Z
| {z }
Qn (ω)
Qn (ω) ≥ n.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
20 / 29
Department for Analysis and Computational Number Theory
The symbolic construction
Iteratively applying the lemma from above, one can construct
the set E :
Γ0 = D∗ ,
[
Γn =
un (ω) Zn · · · Zn (n ≥ 1),
| {z }
ω∈Γn−1
En =
[
Qn (ω)
π([ω]) (n ≥ 0),
ω∈Γn
E=
\
En .
n≥0
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
21 / 29
Department for Analysis and Computational Number Theory
The results
Theorem
I
I
Suppose that every restriction to a part Ti is a
homeomorphism. Then the set of extremely non-normal
numbers is of second category.
Let A be a matrix over Z and D ⊂ Zd /AZd . A tiling T of
Rn is the solution of a set equation of the form
[
A−1 (T + `).
T :=
`∈D
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
22 / 29
Department for Analysis and Computational Number Theory
Outline
Introduction
The non-normal numbers
Normal numbers and fibered systems
The Cantor numeration system
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
23 / 29
Department for Analysis and Computational Number Theory
The Cantor numeration system
Let Q = (qn )∞
n=1 be a sequence of positive integers with qn ≥ 2
for n ≥ 1. For x ∈ [0, 1] the Q-Cantor representation of x is the
(unique) representation of the form
x=
∞
X
n=1
En
q1 q2 · · · qn
where En ∈ {0, 1, . . . , qn − 1} and En 6= 0 infinitely often.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
24 / 29
Department for Analysis and Computational Number Theory
The expectation
Let b be a block of digits of length k. We write Nn (b, x) for the
number of occurrences of the block b in the first n digits of x.
(k)
Furthermore let Qn denote the expectation for the number of
occurrences of a block of length k within the first n digits, i.e.
Qn(k)
=
n
X
j=1
M.G. Madritsch
1
.
qj · · · qj+k−1
Non-normal numbers
CANT, 21 may 2012
25 / 29
Department for Analysis and Computational Number Theory
Two notions of normality
Definition
We call x ∈ [0, 1] k-block-normal if for every block b of length k
we have that
Nn (b, x)
lim
= 1.
(k)
n→∞
Qn
Definition
We call x ∈ [0, 1] distribution normal if the sequence
(qn · · · q1 x)n≥0
is uniformly distributed modulo 1.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
26 / 29
Department for Analysis and Computational Number Theory
Normal and non-normal numbers
For example, the number
x=
∞
X
n=1
En
q1 q2 · · · qn
defined by
En = (1, 1, 2, 1, 2, 3, 1, 2, 3, 4, . . .)
qn = (2, 3, 3, 4, 4, 4, 5, 5, 5, 5, . . .)
is distribution-normal but not block-normal.
Altomare and Mance (2011) constructed a couple sequence of bases
and sequence of digits such that the number x is block-normal but
not distributional-normal.
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
27 / 29
Department for Analysis and Computational Number Theory
µ-normal numbers
Let µ be a probability measure which is shift invariant, i.e. for any
block b we have
X
X
µ(bd) =
µ(db) = µ(b)
d∈D
d∈D
Theorem Madritsch and Mance
Given µ there exists a pair of sequence of digits (En )n≥1 and
sequence of bases (qn )n≥1 such that the number
x=
∞
X
n=1
En
q1 q2 · · · qn
is µ-normal, i.e.
lim
n→∞
M.G. Madritsch
Nn (b, x)
(k)
= µ(b).
Qn
Non-normal numbers
CANT, 21 may 2012
28 / 29
Department for Analysis and Computational Number Theory
2-normal but not 3-normal
30
31
32
33
20
21
22
23
10
11
12
13
00
01
02
03
En = (0, 0, 1, 1, 0, . . .)
qn = (2, 2, 2, 2, 2, . . .)
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
29 / 29
Department for Analysis and Computational Number Theory
2-normal but not 3-normal
30
31
32
33
20
21
22
23
10
11
12
13
00
01
02
03
En = (0, 0, 1, 1, 0, 2, 2, 1, 2, 0, . . .)
qn = (2, 2, 2, 2, 2, 3, 3, 3, 3, 3, . . .)
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
29 / 29
Department for Analysis and Computational Number Theory
2-normal but not 3-normal
30
31
32
33
20
21
22
23
10
11
12
13
00
01
02
03
En = (0, 0, 1, 1, 0, 2, 2, 1, 2, 0, 3, 3, 2, 3, 1, 3, 0, . . .)
qn = (2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, . . .)
M.G. Madritsch
Non-normal numbers
CANT, 21 may 2012
29 / 29