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Normality and Finite State Dimension of Liouville Numbers
Satyadev Nandakumar Santosh Kumar VangepalliIndian Institute of Technology Kanpur
September 2013
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Introduction
⊲ Introduction
Definition
Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
2 / 21
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Normal Numbers
Introduction
⊲ Definition
Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
3 / 21
Definition. [Normal Binary Sequence]
![Page 4: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/4.jpg)
Normal Numbers
Introduction
⊲ Definition
Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
3 / 21
Definition. [Normal Binary Sequence]
An infinite binary sequence ω is said to be normal in base 2 if forevery natural number k and every k-long string w,
limn→∞
|{i ∈ N | ω[i . . . i+ n− 1] = w}|n
=1
2k.
![Page 5: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/5.jpg)
Normal Numbers
Introduction
⊲ Definition
Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
3 / 21
Definition. [Normal Binary Sequence]
An infinite binary sequence ω is said to be normal in base 2 if forevery natural number k and every k-long string w,
limn→∞
|{i ∈ N | ω[i . . . i+ n− 1] = w}|n
=1
2k.
For example, the asymptotic density of 0s is 1/2,
![Page 6: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/6.jpg)
Normal Numbers
Introduction
⊲ Definition
Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
3 / 21
Definition. [Normal Binary Sequence]
An infinite binary sequence ω is said to be normal in base 2 if forevery natural number k and every k-long string w,
limn→∞
|{i ∈ N | ω[i . . . i+ n− 1] = w}|n
=1
2k.
For example, the asymptotic density of 0s is 1/2, the asymptoticdensity of 01 is 1
4 , and so on.
![Page 7: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/7.jpg)
Normal Numbers
Introduction
⊲ Definition
Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
3 / 21
Definition. [Normal Binary Sequence]
An infinite binary sequence ω is said to be normal in base 2 if forevery natural number k and every k-long string w,
limn→∞
|{i ∈ N | ω[i . . . i+ n− 1] = w}|n
=1
2k.
For example, the asymptotic density of 0s is 1/2, the asymptoticdensity of 01 is 1
4 , and so on.
A real number in [0,1] will be called normal if its binaryexpansion is a normal sequence.
![Page 8: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/8.jpg)
Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
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Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
Theorem (Borel, 1909). The set of normal numbers in [0,1] hasLebesgue measure 1.
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Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
Theorem (Borel, 1909). The set of normal numbers in [0,1] hasLebesgue measure 1.
But are there any constructions known?
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Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
Theorem (Borel, 1909). The set of normal numbers in [0,1] hasLebesgue measure 1.
But are there any constructions known?
A simple example of a normal binary sequence:
![Page 12: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/12.jpg)
Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
Theorem (Borel, 1909). The set of normal numbers in [0,1] hasLebesgue measure 1.
But are there any constructions known?
A simple example of a normal binary sequence:
0 1
![Page 13: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/13.jpg)
Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
Theorem (Borel, 1909). The set of normal numbers in [0,1] hasLebesgue measure 1.
But are there any constructions known?
A simple example of a normal binary sequence:
0 1 00 01 10 11
![Page 14: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/14.jpg)
Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
Theorem (Borel, 1909). The set of normal numbers in [0,1] hasLebesgue measure 1.
But are there any constructions known?
A simple example of a normal binary sequence:
0 1 00 01 10 11 000 . . .
![Page 15: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/15.jpg)
Examples of Normal Numbers
Introduction
Definition
⊲ Examples
Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
4 / 21
Are there normal numbers in [0,1]?
Theorem (Borel, 1909). The set of normal numbers in [0,1] hasLebesgue measure 1.
But are there any constructions known?
A simple example of a normal binary sequence:
0 1 00 01 10 11 000 . . .
This is known as the Champernowne sequence (1933).
![Page 16: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/16.jpg)
Properties of Normal Numbers
Introduction
Definition
Examples
⊲ Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
5 / 21
No normal number can be rational.
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Properties of Normal Numbers
Introduction
Definition
Examples
⊲ Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
5 / 21
No normal number can be rational.
Definition. A real number is algebraic if it is the root of apolynomial with integer coefficients. e.g.
√2 is the root of the
polynomial x2 − 2.
![Page 18: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/18.jpg)
Properties of Normal Numbers
Introduction
Definition
Examples
⊲ Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
5 / 21
No normal number can be rational.
Definition. A real number is algebraic if it is the root of apolynomial with integer coefficients. e.g.
√2 is the root of the
polynomial x2 − 2.Non-algebraic irrational numbers are called transcendental. (e.g.π, e etc.)
![Page 19: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/19.jpg)
Properties of Normal Numbers
Introduction
Definition
Examples
⊲ Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
5 / 21
No normal number can be rational.
Definition. A real number is algebraic if it is the root of apolynomial with integer coefficients. e.g.
√2 is the root of the
polynomial x2 − 2.Non-algebraic irrational numbers are called transcendental. (e.g.π, e etc.)
Are normal numbers always transcendental? (We do not know)
![Page 20: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/20.jpg)
Properties of Normal Numbers
Introduction
Definition
Examples
⊲ Properties
Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
5 / 21
No normal number can be rational.
Definition. A real number is algebraic if it is the root of apolynomial with integer coefficients. e.g.
√2 is the root of the
polynomial x2 − 2.Non-algebraic irrational numbers are called transcendental. (e.g.π, e etc.)
Are normal numbers always transcendental? (We do not know)
Can we construct normal numbers which are provablytranscendental? What can we say about normal numbers inspecial classes of transcendental numbers?
![Page 21: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/21.jpg)
Transcendence and Normality
Introduction
Definition
Examples
Properties
⊲ Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
6 / 21
Champernowne number is a transcendental number. (Mahler,1937)
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Transcendence and Normality
Introduction
Definition
Examples
Properties
⊲ Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
6 / 21
Champernowne number is a transcendental number. (Mahler,1937)
Some transcendental numbers are far from normal.
Example. Liouville constant1 defined as
α =∞∑
i=0
1
2i!.
is a transcendental number.
![Page 23: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/23.jpg)
Transcendence and Normality
Introduction
Definition
Examples
Properties
⊲ Transcendence
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
6 / 21
Champernowne number is a transcendental number. (Mahler,1937)
Some transcendental numbers are far from normal.
Example. Liouville constant1 defined as
α =∞∑
i=0
1
2i!.
is a transcendental number.
α cannot be normal - e.g. 111 never appears in α.
1in base 2
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Normality and Diophantine Approximation
Introduction
⊲
Normality andDiophantineApproximation
Liouville Numbers
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
7 / 21
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Liouville Numbers
8 / 21
Theorem (Liouville, 1854). A real number γ is an algebraic number of degreed. Then there is a constant C such that for every pair of natural numbers aand b,
∣
∣
∣γ − a
b
∣
∣
∣<
C
bd.
![Page 26: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/26.jpg)
Liouville Numbers
8 / 21
Theorem (Liouville, 1854). A real number γ is an algebraic number of degreed. Then there is a constant C such that for every pair of natural numbers aand b,
∣
∣
∣γ − a
b
∣
∣
∣<
C
bd.
We can define the following class of transcendentals.
![Page 27: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/27.jpg)
Liouville Numbers
8 / 21
Theorem (Liouville, 1854). A real number γ is an algebraic number of degreed. Then there is a constant C such that for every pair of natural numbers aand b,
∣
∣
∣γ − a
b
∣
∣
∣<
C
bd.
We can define the following class of transcendentals.
Definition. A real number β is called a Liouville Number if there is a
sequence of rationals(
piqi
)
∞
i=1such that
∣
∣
∣
∣
β − piqi
∣
∣
∣
∣
≤ 1
qii.
![Page 28: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/28.jpg)
Liouville Numbers
8 / 21
Theorem (Liouville, 1854). A real number γ is an algebraic number of degreed. Then there is a constant C such that for every pair of natural numbers aand b,
∣
∣
∣γ − a
b
∣
∣
∣<
C
bd.
We can define the following class of transcendentals.
Definition. A real number β is called a Liouville Number if there is a
sequence of rationals(
piqi
)
∞
i=1such that
∣
∣
∣
∣
β − piqi
∣
∣
∣
∣
≤ 1
qii.
Liouville’s constant is a Liouville number. (first number provedtranscendental).
![Page 29: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/29.jpg)
Normality of Liouville numbers
Introduction
Normality andDiophantineApproximation
⊲Normality ofLiouville numbers
Question
de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
9 / 21
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Normality of Liouville Numbers
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
⊲ Question
de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
10 / 21
Liouville numbers have Lebesgue measure 0. (e.g. Oxtoby, 1980)
![Page 31: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/31.jpg)
Normality of Liouville Numbers
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
⊲ Question
de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
10 / 21
Liouville numbers have Lebesgue measure 0. (e.g. Oxtoby, 1980)They even have Hausdorff dimension 0. (e.g. Oxtoby, 1980).Further, they have constructive Hausdorff Dimension 0. (Staiger,2002)
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Normality of Liouville Numbers
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
⊲ Question
de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
10 / 21
Liouville numbers have Lebesgue measure 0. (e.g. Oxtoby, 1980)They even have Hausdorff dimension 0. (e.g. Oxtoby, 1980).Further, they have constructive Hausdorff Dimension 0. (Staiger,2002)
No easy measure-theoretic existence proof. (A measure 0 setmay or may not meet a measure 1 set.)
![Page 33: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/33.jpg)
Normality of Liouville Numbers
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
⊲ Question
de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
10 / 21
Liouville numbers have Lebesgue measure 0. (e.g. Oxtoby, 1980)They even have Hausdorff dimension 0. (e.g. Oxtoby, 1980).Further, they have constructive Hausdorff Dimension 0. (Staiger,2002)
No easy measure-theoretic existence proof. (A measure 0 setmay or may not meet a measure 1 set.)
Liouville’s constant is non-normal.
![Page 34: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/34.jpg)
Normality of Liouville Numbers
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
⊲ Question
de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
10 / 21
Liouville numbers have Lebesgue measure 0. (e.g. Oxtoby, 1980)They even have Hausdorff dimension 0. (e.g. Oxtoby, 1980).Further, they have constructive Hausdorff Dimension 0. (Staiger,2002)
No easy measure-theoretic existence proof. (A measure 0 setmay or may not meet a measure 1 set.)
Liouville’s constant is non-normal.
Surprisingly, there are normal Liouville numbers.
![Page 35: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/35.jpg)
Idea: de Bruijn Strings
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
⊲ de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
11 / 21
A binary string w of length 2n is called a de-Bruijn string oforder n if every n length string occurs exactly once in w, whenlooked in a cyclic manner.
![Page 36: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/36.jpg)
Idea: de Bruijn Strings
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
⊲ de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
11 / 21
A binary string w of length 2n is called a de-Bruijn string oforder n if every n length string occurs exactly once in w, whenlooked in a cyclic manner.
e.g. 0110 is a de-Bruin sequence of order 2:
0110
![Page 37: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/37.jpg)
Idea: de Bruijn Strings
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
⊲ de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
11 / 21
A binary string w of length 2n is called a de-Bruijn string oforder n if every n length string occurs exactly once in w, whenlooked in a cyclic manner.
e.g. 0110 is a de-Bruin sequence of order 2:
0110 0110
![Page 38: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/38.jpg)
Idea: de Bruijn Strings
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
⊲ de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
11 / 21
A binary string w of length 2n is called a de-Bruijn string oforder n if every n length string occurs exactly once in w, whenlooked in a cyclic manner.
e.g. 0110 is a de-Bruin sequence of order 2:
0110 0110 0110
![Page 39: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/39.jpg)
Idea: de Bruijn Strings
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
⊲ de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
11 / 21
A binary string w of length 2n is called a de-Bruijn string oforder n if every n length string occurs exactly once in w, whenlooked in a cyclic manner.
e.g. 0110 is a de-Bruin sequence of order 2:
0110 0110 0110 0110
![Page 40: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/40.jpg)
Idea: de Bruijn Strings
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
⊲ de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
11 / 21
A binary string w of length 2n is called a de-Bruijn string oforder n if every n length string occurs exactly once in w, whenlooked in a cyclic manner.
e.g. 0110 is a de-Bruin sequence of order 2:
0110 0110 0110 0110
Theorem (N. G. de Bruijn 1946, I. J. Good 1946). Let Σ be afinite alphabet, and k ∈ N. Then there exists a de Bruijn stringfrom the alphabet Σ of order k. This string has length |Σ|k.
![Page 41: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/41.jpg)
Idea: de Bruijn Strings
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
⊲ de Bruijn strings
Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
11 / 21
A binary string w of length 2n is called a de-Bruijn string oforder n if every n length string occurs exactly once in w, whenlooked in a cyclic manner.
e.g. 0110 is a de-Bruin sequence of order 2:
0110 0110 0110 0110
Theorem (N. G. de Bruijn 1946, I. J. Good 1946). Let Σ be afinite alphabet, and k ∈ N. Then there exists a de Bruijn stringfrom the alphabet Σ of order k. This string has length |Σ|k.
de Bruijn sequences are a standard tool in the study of normality.
![Page 42: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/42.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
![Page 43: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/43.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
β =
![Page 44: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/44.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
β = B(1)11
![Page 45: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/45.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
β = B(1)11
B(2)22
![Page 46: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/46.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
β = B(1)11
B(2)22
. . . B(k)kk
. . .
![Page 47: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/47.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
β = B(1)11
B(2)22
. . . B(k)kk
. . .
β is a normal Liouville number.
![Page 48: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/48.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
β = B(1)11
B(2)22
. . . B(k)kk
. . .
β is a normal Liouville number.
Intuition (Liouville Number): The stages have lengthsapproximately k! as in Liouville’s constant. Hence we can try toform a diophantine approximation as in Liouville’s constant.
![Page 49: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/49.jpg)
Construction
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
⊲ Construction
Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
12 / 21
Denote the lexicographically least binary de Bruijn string of orderk as B(k). Consider the number defined as follows.
β = B(1)11
B(2)22
. . . B(k)kk
. . .
β is a normal Liouville number.
Intuition (Liouville Number): The stages have lengthsapproximately k! as in Liouville’s constant. Hence we can try toform a diophantine approximation as in Liouville’s constant.
Intution (Normality): The basic building block of each stage k isa balanced string of all orders ≤ k.
![Page 50: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/50.jpg)
β is a Liouville number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
⊲ Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
13 / 21
The length of stage k is 2k kk.
![Page 51: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/51.jpg)
β is a Liouville number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
⊲ Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
13 / 21
The length of stage k is 2k kk. The length of the prefix up toand including stage k is Lk =
∑ki=0 2
iii = O((2k)k).
![Page 52: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/52.jpg)
β is a Liouville number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
⊲ Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
13 / 21
The length of stage k is 2k kk. The length of the prefix up toand including stage k is Lk =
∑ki=0 2
iii = O((2k)k).
Consider the rational number bk at level k defined to have thesame binary expansion as β up to stage k, followed by a periodicpattern of B(k + 1).
![Page 53: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/53.jpg)
β is a Liouville number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
⊲ Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
13 / 21
The length of stage k is 2k kk. The length of the prefix up toand including stage k is Lk =
∑ki=0 2
iii = O((2k)k).
Consider the rational number bk at level k defined to have thesame binary expansion as β up to stage k, followed by a periodicpattern of B(k + 1). The length of its denominator is O(Lk).
![Page 54: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/54.jpg)
β is a Liouville number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
⊲ Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
13 / 21
The length of stage k is 2k kk. The length of the prefix up toand including stage k is Lk =
∑ki=0 2
iii = O((2k)k).
Consider the rational number bk at level k defined to have thesame binary expansion as β up to stage k, followed by a periodicpattern of B(k + 1). The length of its denominator is O(Lk).
|bk − β| ≤ 1
2(2(k+1))k+1≤ 1
(2O((2k)k))k
![Page 55: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/55.jpg)
β is a Liouville number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
⊲ Liouvilleness
Normality
Generalization:Finite StateDimension
Open Questions
13 / 21
The length of stage k is 2k kk. The length of the prefix up toand including stage k is Lk =
∑ki=0 2
iii = O((2k)k).
Consider the rational number bk at level k defined to have thesame binary expansion as β up to stage k, followed by a periodicpattern of B(k + 1). The length of its denominator is O(Lk).
|bk − β| ≤ 1
2(2(k+1))k+1≤ 1
(2O((2k)k))k
Hence β is a Liouville number.
![Page 56: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/56.jpg)
β is a normal number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
Liouvilleness
⊲ Normality
Generalization:Finite StateDimension
Open Questions
14 / 21
In B(k), every string of length k appears once (wrappingaround).
![Page 57: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/57.jpg)
β is a normal number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
Liouvilleness
⊲ Normality
Generalization:Finite StateDimension
Open Questions
14 / 21
In B(k), every string of length k appears once (wrappingaround). Hence every string w of length n ≤ k appears with theright frequency 1
2k−n .
![Page 58: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/58.jpg)
β is a normal number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
Liouvilleness
⊲ Normality
Generalization:Finite StateDimension
Open Questions
14 / 21
In B(k), every string of length k appears once (wrappingaround). Hence every string w of length n ≤ k appears with theright frequency 1
2k−n .
Hence for all k ≥ |w|, the frequency of any string w is 12w at the
end of every block B(k).
![Page 59: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/59.jpg)
β is a normal number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
Liouvilleness
⊲ Normality
Generalization:Finite StateDimension
Open Questions
14 / 21
In B(k), every string of length k appears once (wrappingaround). Hence every string w of length n ≤ k appears with theright frequency 1
2k−n .
Hence for all k ≥ |w|, the frequency of any string w is 12w at the
end of every block B(k). What about within B(k)?
![Page 60: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/60.jpg)
β is a normal number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
Liouvilleness
⊲ Normality
Generalization:Finite StateDimension
Open Questions
14 / 21
In B(k), every string of length k appears once (wrappingaround). Hence every string w of length n ≤ k appears with theright frequency 1
2k−n .
Hence for all k ≥ |w|, the frequency of any string w is 12w at the
end of every block B(k). What about within B(k)?
For every large enough k, length(B(k)) = o(Lk−1).
![Page 61: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/61.jpg)
β is a normal number
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Question
de Bruijn strings
Construction
Liouvilleness
⊲ Normality
Generalization:Finite StateDimension
Open Questions
14 / 21
In B(k), every string of length k appears once (wrappingaround). Hence every string w of length n ≤ k appears with theright frequency 1
2k−n .
Hence for all k ≥ |w|, the frequency of any string w is 12w at the
end of every block B(k). What about within B(k)?
For every large enough k, length(B(k)) = o(Lk−1). Hence thedeviations from balance within B(k) is insignificant for lengthsfrom Lk−1 to Lk−1 + length(Bk).
![Page 62: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/62.jpg)
Generalization: Finite State Dimension
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
⊲
Generalization:Finite StateDimension
Definition
Normality and FSD
Construction
Open Questions
15 / 21
![Page 63: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/63.jpg)
Finite-State Dimension
16 / 21
Consider the n-long prefix of an infinite binary sequence ω. Then for k < n,the frequency of a k-long word w in ω[0 . . . n− 1] is
π(ω[0 . . . n− 1], w) =|{i | 0 ≤ i ≤ n− k − 1 ω[i . . . i+ k − 1] = w}|
n− k − 1.
![Page 64: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/64.jpg)
Finite-State Dimension
16 / 21
Consider the n-long prefix of an infinite binary sequence ω. Then for k < n,the frequency of a k-long word w in ω[0 . . . n− 1] is
π(ω[0 . . . n− 1], w) =|{i | 0 ≤ i ≤ n− k − 1 ω[i . . . i+ k − 1] = w}|
n− k − 1.
The entropy of this probability distribution is
Hk,n(ω) =∑
w∈Σk
−π(ω[0 . . . n− 1], w) log π(ω[0 . . . n− 1], w).
![Page 65: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/65.jpg)
Finite-State Dimension
16 / 21
Consider the n-long prefix of an infinite binary sequence ω. Then for k < n,the frequency of a k-long word w in ω[0 . . . n− 1] is
π(ω[0 . . . n− 1], w) =|{i | 0 ≤ i ≤ n− k − 1 ω[i . . . i+ k − 1] = w}|
n− k − 1.
The entropy of this probability distribution is
Hk,n(ω) =∑
w∈Σk
−π(ω[0 . . . n− 1], w) log π(ω[0 . . . n− 1], w).
The k-entropy rate of ω is defined as
Hk(ω) =1
klim infn→∞
Hk,n(ω).
![Page 66: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/66.jpg)
Finite-State Dimension
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Consider the n-long prefix of an infinite binary sequence ω. Then for k < n,the frequency of a k-long word w in ω[0 . . . n− 1] is
π(ω[0 . . . n− 1], w) =|{i | 0 ≤ i ≤ n− k − 1 ω[i . . . i+ k − 1] = w}|
n− k − 1.
The entropy of this probability distribution is
Hk,n(ω) =∑
w∈Σk
−π(ω[0 . . . n− 1], w) log π(ω[0 . . . n− 1], w).
The k-entropy rate of ω is defined as
Hk(ω) =1
klim infn→∞
Hk,n(ω).
The finite-dimensional entropy rate of ω is defined as H(ω) = infk Hk(ω).
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Normality and Finite-state dimension
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Definition
⊲Normality andFSD
Construction
Open Questions
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A binary sequence is normal if and only if it has finite statedimension 1 (Schnorr and Stimm 1971,
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Normality and Finite-state dimension
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Definition
⊲Normality andFSD
Construction
Open Questions
17 / 21
A binary sequence is normal if and only if it has finite statedimension 1 (Schnorr and Stimm 1971, Becher and Heiber 2012.)
![Page 69: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/69.jpg)
Normality and Finite-state dimension
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Definition
⊲Normality andFSD
Construction
Open Questions
17 / 21
A binary sequence is normal if and only if it has finite statedimension 1 (Schnorr and Stimm 1971, Becher and Heiber 2012.)
Every binary sequence has a finite-state dimension between 0and 1.
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Normality and Finite-state dimension
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Definition
⊲Normality andFSD
Construction
Open Questions
17 / 21
A binary sequence is normal if and only if it has finite statedimension 1 (Schnorr and Stimm 1971, Becher and Heiber 2012.)
Every binary sequence has a finite-state dimension between 0and 1.
Finite-state dimension thus provides a quantification of the“non-normality” of binary expansions of reals.
![Page 71: Satyadev Nandakumar Santosh Kumar Vangepalli Indian ...PDF/FSDNLN.pdf · 1 / 21 Normality and Finite State Dimension of Liouville Numbers Satyadev Nandakumar Santosh Kumar Vangepalli](https://reader035.vdocuments.site/reader035/viewer/2022063005/5fb5048befcd5449f415286b/html5/thumbnails/71.jpg)
Normality and Finite-state dimension
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Definition
⊲Normality andFSD
Construction
Open Questions
17 / 21
A binary sequence is normal if and only if it has finite statedimension 1 (Schnorr and Stimm 1971, Becher and Heiber 2012.)
Every binary sequence has a finite-state dimension between 0and 1.
Finite-state dimension thus provides a quantification of the“non-normality” of binary expansions of reals.
Can we produce Liouville numbers of any given finite-statedimension in [0, 1]?
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Liouville numbers of any FSD between 0 and 1
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Let m,n be positive integers, m < n. We construct a Liouville number withfinite-state dimension m/n.
αm/n = 0·(
(021
)n−mB(1)m)11 (
(022
)n−mB(2)m)22
. . .(
(02k
)n−mB(k)m)kk
.
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Liouville numbers of any FSD between 0 and 1
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Let m,n be positive integers, m < n. We construct a Liouville number withfinite-state dimension m/n.
αm/n = 0·(
(021
)n−mB(1)m)11 (
(022
)n−mB(2)m)22
. . .(
(02k
)n−mB(k)m)kk
.
αm/n is a Liouville number with finite-state dimension m/n.
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Open Questions
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
⊲ Open Questions
Open Questions
.
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Open Questions
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
⊲ Open Questions
.
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� Can we construct absolutely normal Liouville numbers? Theyexist. (Bugeaud, 2002)
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Open Questions
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
⊲ Open Questions
.
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� Can we construct absolutely normal Liouville numbers? Theyexist. (Bugeaud, 2002)
� Can we construct algebraic normal numbers?
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.
Introduction
Normality andDiophantineApproximation
Normality ofLiouville numbers
Generalization:Finite StateDimension
Open Questions
Open Questions
⊲ .
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Thank You!