RATIONAL WALKS

A Thesis

Presented to the

Faculty of

California State Polytechnic University, Pomona

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science

In

Mathematics

By

Anthony Simon

2019 SIGNATURE PAGE

THESIS: RATIONAL WALKS

AUTHOR: Anthony Simon

DATE SUBMITTED: Spring 2019

Department of Mathematics and Statistics

Dr. Randall Swift Thesis Committee Chair Mathematics & Statistics

Dr. Michael Green Mathematics & Statistics

Dr. Alan Krinik Mathematics & Statistics

ii ACKNOWLEDGMENTS

I primarily thank all my roommates that I have lived with for the past years, there has been so much growth in everyone it encourages me to keep doing great things for the world. I also like to thank Dr. Swift for finding time to read my drafts, I know the transition from quarters to semesters for Cal Poly has been rough but, this is pivotal for the schools growth.

iii ABSTRACT

In this thesis a mapping from decimals .d1d2...dn... ∈ [0, 1] into {0, 1, 2} is defined and described as a walk. The thesis explores visual representations of rational and irrational numbers as walks. Several interesting classes of real numbers are considered, including normal numbers and Khinchin’s Constant.

iv Contents

Signature Page ii

Acknowledgements iii

Abstract iv

List of Figures viii

List of Tables viii

Chapter 1 Introduction 1

1.1 Sets of Real Numbers ...... 1

1.2 Ternary Strings ...... 6

1.3 Rational Mapping ...... 7

1.3.1 Examples of Rational Maps ...... 8

Chapter 2 Arithmetic and Geometric Operations on Strings 12

2.1 Operations Arithmetic ...... 12

2.2 Substrings ...... 19

2.3 Some Walks ...... 22

2.4 String Analysis ...... 24

v 2.5 Characterizations of Strings ...... 27

Chapter 3 Analysis of Normal Strings 31

3.1 Normal Numbers ...... 31

3.2 Normal Strings ...... 33

Bibliography 36

Appendices 37

Appendix A Code 37

vi List of Figures

1.1 The Rational Map of .22439 ...... 8

1.2 The Rational Map of .27388 ...... 9

1.3 The Rational Map of .53850 ...... 10

1.4 The Rational Map of .57845 ...... 11

2.1 The Rational Map of 1 ...... 21 27 [3,7] 2.2 The Rational Map of 1 ...... 22 7 [1,30] √ 2.3 The Rational Map of 2[1,100] ...... 22

2.4 The Rational Map of π[1,100] ...... 23

2.5 The Rational Map of π[1,1000] ...... 23

2.6 The Rational Map of Champernowne(10)[1,48] ...... 28 2.7 The Rational Map of Khinchin’s constant first 100 decimals . . . . 30

2.8 The Rational Map of Khinchin’s constant first 1,000 decimals . . . 30

3.1 The First 1,000 Steps of Champernowne(10) ...... 33

3.2 The First 100,000 Steps of Champernowne(10) ...... 33

3.3 The First 1000 Steps of the Copland-Erdos Constant ...... 34

3.4 The First 4000 Steps of the Copland-Erdos Constant ...... 34

3.5 The First 8000 Steps of the Copland-Erdos Constant ...... 35

3.6 The First 25000 Steps of the Copland-Erdos Constant ...... 35

vii List of Tables

2.1 2-step walks ...... 18

viii Chapter 1

Introduction

In this chapter the familiar sets of natural numbers, integers, rational numbers and

irrational numbers are introduced. A class named the ternary class composed of equivalence classes named strings. Finally, there will be a definition of how to plot

these strings.

1.1 Sets of Real Numbers

Some sets of numbers that are used in this thesis are detailed here.

Definition The set of natural numbers, N, is defined as

N = {1, 2, 3, ...}.

An alternative name for the natural numbers is the counting numbers.

Notation: The natural numbers including zero will be denoted as, N0 = N ∪ {0}, and is sometimes called the whole numbers.

Definition The set of integers, Z, is

Z = {..., −2, −1, 0, 1, 2, ...}.

1 Definition For n ∈ N and a, b ∈ Z, a and b are congruent modulo n, if their difference a − b is an integer multiple of n namely, if there is an integer k such that

a − b = kn. This is denoted

a ≡ b (mod n).

Notation The set of integers modulo n, Zn, is defined as

Zn = {0, 1, 2, ..., n − 1},

where n ∈ N.

In general this thesis focuses on the set Z3 = {0, 1, 2}. If Z2 = {0, 1} is the binary class, then Z3 is called the ternary class.

Definition The set of rational numbers, Q is defined as follows,

a = { |a, b ∈ , b =6 0, gcd(a, b) = 1} Q b Z

In the definition the constraint gcd(a, b) = 1 gives that the representation of numbers in Q are in lowest form.

11 31 For instance, the number 0.5 could be expressed as 22 , 62 , or any other fraction but 1 with gcd(a, b) = 1, the only fraction that is appropriate is 2 . To define the mantissa of a number, there needs to be a definition of the floor function, since the only numbers of intrest are in the interval (0, 1).

Definition The floor function, bxc is the greatest integer les than x.

Examples

bπc = 3

bec = 2

2 The mantissa is the positive fraction part of a number.

Definition The mantissa of a real number x is

x − bxc = 0.x1x2...,

where each xi ∈ Zb, where b ∈ N − {1} is a base. We consider a decimal expansion that either terminates or repeats.

Example

What is the fraction representation of the number .333...?

3 3 .3333... = + + ... 10 100 ∞ X 3 = 10n+1 n=0 3 10 = 1 1 − 10 3 10 = ∗ 10 9 1 = 3

Lets generalize this idea from base 10 to base b, where b ∈ N − {1}. Theorem Every finite repeating decimal number is a rational number.

Proof. For an x ∈ (0, 1)

x = .d1d2...dnd1d2...(base b)

where each di ∈ Zb, where b ∈ N − {1}. Since this number can be written in the

3 form

x = .d1d2...dn−1dnd1d2...(base b) d d d d d d d d d = 1 +2 + ... + n−1 +n +1 +2 + ... + n−1 +n + 1 + ... b b2 bn−1 bn bn+1 bn+2 b2n−1 b2 n b2n+1 d d d d d d = (1 +1 + 1 + ...) + ( 2 +2 + 2 + ...) + b bn+1 b2n+1 b2 bn+2 b2n+2 d d d d d d ... +(n−1 + n−1 + n−1 + ...) + ( n + n +n + ...) bn−1 b2n−1 b3n−1 bn b2n b3 d 1 1 d 1 1 d 1 1 =1 (1 + + + ...) + 2 (1 + + + ...) + ... +n (1 + + + ...) b bn b2n b2 bn b2n bn bn b2n

Represented by using the sum of a geometric series

1 1 d d d d = (1 + + + ...)(1 + 2 + ... + n−1 +n ) bn b2n b b2 bn−1 bn 1 d bn−1 + d bn−2 + ... + d b + d = ( )( 1 2 n−1 n ) 1 n 1 − b n b bn d bn−1 + d bn−2 + ... + d b + d d bn−1 + d bn−2 + ... + d b + d = ( )( 1 2 n−1 n ) = 1 2 n−1 n bn − 1 bn bn − 1

n−1 n−2 n Since d1b + d2b + ... + dn−1b + dn is an integer and b − 1 is also an integer then x is rational. □

4 Example

The number x ∈ (0, 1) is a repeating base 6 decimal, for x = .1234512345...(base 6) the fraction representation is this number.

x = .12345...12345...(base 6) (1)64 + (2)63 + (3)62 + (4)6 + 5 = 65 − 1 1865 373 = = . 7775 1555

Finally the ternary string class will be defined.

Definition Let T be the set

T = { x | x = [δ1 δ2 ...], δi = Z3} given, x = .d1d2... each di ∈ Zb, where b ∈ N/{1}. Notation: A ternary string will be denoted in this thesis as string x.

Example

21 Show 25 = [2 1]. 21 Write 25 as a decimal 21 = .84. 25

Now since d1 = 8 and d2 = 4, the corresponding δi’s are, δ1 = 8 (mod 3) = 2 and

21 21 δ2 = 4 (mod 3) = 1. So the string of 25 is [2 1]. Thus 25 = [2 1].

Notation We denote finite repeating rational classes by [δ1 δ2 ... δn] Examples

Show 1/3 = [0]

Begin with, converting 1/3 to it’s decimal form

1 = .333... 3

5 meaning each di = 3, thus each δi = 3 (mod 3) = 0. So, the string of 1/3 is

[0 0 0 0 0 ...] = [0]

1 Thus 3 = [0].

1.2 Ternary Strings

In this section we consider the structure of the ternary class, T, this is done by defining size and length of a string.

Definition The length of a string [d1 ... dn] is

l([d1 ... dn]) = n.

If a string does not terminate, it is said to be of unbounded length.

Since 4 digits from Z10 map to 0 ∈ Z3, and 3 digits from Z10 map to 1, 2 ∈ Z3. Definition The size of the string x is defined as

n0 n1+n2 s(x) = s([δ1...δn]) = (4) (3) ,

where n0 is the number of δi’s that are 0, n1 is the number of δi’s that are 1 and

n2 is the number of δi’s that are 2. The size of a string is how many numbers from (0, 1) map to an individual string

in T. Note: A string of unbounded length has unbounded size.

Example

Determine the size of each string [2 0 0 1], [2 0 2 0] and [0].

s([2 0 0 1]) = (4)2(3)1+1 = 144,

6 notice that s([1 0 2 0]) = 144 as well.

Note: The size of the class is dependent on the number of zeros.

Since [0] has an unbounded length then s([0]) is of unbounded size since,

s([0]) = lim 4n = ∞. n→∞

1.3 Rational Mapping

We will now consider a graphical method to visualize a rational number. For the

ternary class, each δi ∈ {0, 1, 2} = Z3, so we consider a mapping for each string by

assuming a direction to each digit 0, 1 or 2. Each δi ∈ Z3 is mapped to one of three

th directions, straight, up or down. If the i digit δi is a zero, the map stays at the same height. If δi is 1, the map assigns an upward step of height 1. If δi is 2 the map assigns a downward step of size 1.

Definition Each δi ∈ Z3 is mapped to a direction: ⎧ ⎪ ⎪0 δi = 0 ⎪ ⎨⎪ Di = 1 δi = 1 ⎪ ⎪ ⎪ ⎩⎪−1 δi = 2

A rational mapping of a finite string x, is the collection of steps

Sn = {D1,D2, ..., Dn}.

Since the strings from irrational numbers and rational repeating numbers have

unbounded length, their corresponding strings cannot be plotted. On the next

couple of pages there will be the rational walks of a few finite length strings.

7 1.3.1 Examples of Rational Maps

1. For the decimal 0.22439

d1 = d2 = 2,

so that

δ1 = δ2 = 2,

which means that the first two steps of the graph are down. Then δ3 = 1 as

d3 = 4 so the next step is up. Finally since

δ4 = δ5 = 0, since d4 = 3, d5 = 9

ending on two straight segments. Thus the string is [2 2 1 0 0]. The graph of

this is shown in Figure 1.1.

1 2 3 4 5

-0.5

-1.0

-1.5

-2.0

Figure 1.1: The Rational Map of .22439

8 2. For the decimal 0.27388 since d1 = 2, δ1 = 2, which means that the first

step of the graph is down. Then d2 = 7, so δ2 = 7 (mod 3) = 1 so the next

step is up. Next, d3 = 3, so δ3 = 3 (mod 3) = 0, which is a step straight. Finally

d4 = d5 = 8,

thus,

δ4 = δ5 = 8 (mod 3) = 2,

ending on two down segments. Thus the string is [2 1 0 2 2]. This graph is

shown in Figure 1.2.

1 2 3 4 5

-0.5

-1.0

-1.5

-2.0

Figure 1.2: The Rational Map of .27388

9 3. For the decimal 0.53850 the string is [2 0 2 2 0]. Graphically this string

only takes straight steps and downward steps. This graph is shown in Figure

1.3.

1 2 3 4 5

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

Figure 1.3: The Rational Map of .53850

10 4. For the decimal 0.57845 the string is [2 1 2 1 2]. Notice though that

δ1 = δ3 = δ5 = 2,

and

δ2 = δ4 = 1,

Graphically this string takes on an interesting pattern. This graph is shown

in Figure 1.4.

-0.2 1 2 3 4 5 -0.4 -0.6 -0.8 -1.0

Figure 1.4: The Rational Map of .57845

Can these graphical representations help us understand what it means to be rational or irrational? That is the question we study next.

11 Chapter 2

Arithmetic and Geometric

Operations on Strings

In this chapter operations on strings will be considered. The operations developed will be helpful in the subsequent analysis of rational maps.

2.1 Operations Arithmetic

These definitions are used to later analyze the compositions of the strings. The

operation of addition can be extended from Z3 to T.

Definition For strings x = [δ1 ... δn] and y = [1 ... m], where each δi, i ∈ Z3 and 1 ≤ i ≤ n ≤ m, the sum is:

x + y = [δ1 + 1 ... δn + n ... 0 + m], where each δi + i ∈ Z3, the sum is component wise mod 3. -- The string [0 ... 0] = [0] is called the zero string of T.

12 The string [−δ1 ... − δn] is called the additive inverse of string [δ1 ... δn], and

[δ1 ... δn] − [1 ... m] = [δ1 ... δn] + [−1 ... − m], defines subtraction.

Examples

1. Compute

[0 1 2] + [2 1 0] and [2 1 0] + [0 1 2]

As follows,

[0 1 2] + [2 1 0] = [(0 + 2) (1 + 1) (2 + 0)]

= [2 2 2]

and

[2 1 0] + [0 1 2] = [(2 + 0) (1 + 1) (0 + 2)]

= [2 2 2]

Proposition String addition is commutative.

Proof. Let x = [δ1 ... δn] and y = [1 ... n] be two strings. Then

x + y = [(δ1 + 1) ... (δn + n)]

= [(1 + δ1) ... (n + δn)] = y + x

□

13 2. The string lengths do not have to be the same. An example will be

provided,

[0 1 2] + [2] = [0 1 2] + [2 0 0]

= [(0 + 2) (1 + 0) (2 + 0)]

= [2 1 2]

String subtraction uses the idea that

−1 (mod 3) = 2,

and

−2 (mod 3) = 1,

so, for example

[2 2 2] − [0 1 2] = [2 2 2] + [0 − 1 − 2]

= [2 2 2] + [0 2 1]

= [(2 + 0) (2 + 2) (2 + 1)] = [2 1 0]

Definition For strings x = [δ1 ... δn] and y = [1 ... m], where each δi, i ∈ Z3 and 1 ≤ i ≤ n ≤ m, the difference is:

x − y = [δ1 − 1 ... δn − n ... 0 − m], where each δi − i ∈ Z3.

14 Definition For two strings x = [δ1 ... δn] and y = [1 ... m], where each δi, i ∈ Z3 and 1 ≤ i ≤ n ≤ m, the product is

x × y = [δ11 ... δnn 0n+1 0n+2 ... 0m]

where each δi ∗ i ∈ Z3. Definition If we take the mth power of the string string x it is

m m m m m x = [δ1 δ2 ... δn] = [δ1δ 2 ... δn ],

where m ∈ N. Examples

1. When multiplying notice the operation is element wise,

[1 1 1 1] × [1 0 1 2] = [(1 ∗ 1) (1 ∗ 0) (1 ∗ 1) (1 ∗ 2)]

= [1 0 1 2]

2. Notice two strings do not have to be the same length in order to perform

string multiplication

[1 0 1 2] × [0 1 2] = [(1 ∗ 0) (0 ∗ 1) (1 ∗ 2) (2 ∗ 0)]

= [0 0 2 0]

3. Squaring strings is as follows,

[0 1 2]2 = [(0 ∗ 0) (1 ∗ 1) (2 ∗ 2)]

= [0 1 1]

3 Notice, for each δi ∈ Z3 that δi = δi, if δi = 0 or δi = 1 this is obvious, but if δi = 2

3 then δ1 = 8 (mod 3) = 2 = δi. Proposition Let x be a string then x = x3 .

15 Proof. Let x be a string. Then x = [δ1 δ2 ...], where each δi ∈ Z3. Consider

3 3 3 x = [δ1 δ2 ...] = [δ1 δ2 ...] = x.

3 Thus x = x. □

Having a tool to sum up the δi’s in a given string will be helpful for analyzing that string later on.

Definition The total weight of our string x is defined by

n X ||x|| = ||[d1 d2 ... dn]|| = δi i=1 Examples

Find the totals of the strings.

||[0 1 2]|| = 0 + 1 + 2 = 3

||[0 1 2 2] − [0 1 2 2]2|| = ||[0 0 1 1]|| = 2

||[0]|| = 0

We can concatenate strings together, to form larger classes contained in the ternary class, T. The operation proceeds as follows.

Definition For strings x = [δ1 ... δn] and y = [1 ... m], where each δi, i ∈ Z3 and 1 ≤ i ≤ n ≤ m, the concatenation is

x ⊕ y = [δ1 ... δn 1 ... m].

16 Example

What is the string of 1/6?

Writing 1/6 as a decimal 0.16666.... Since d1 = 1 and all other di = 6, where i > 1, we have δ1 = 1 and δi = 0, where i > 1. Thus the string

-- 1/6 = [1] ⊕ [0].

Using string concatenation any finite length string can be built from strings of length 2. In Table 2.1 all the possible length 2 strings are listed and the decimals that equate to them. All the figures in Chapter 1 could be built using elements from table 2.1.

17 Table 2.1: All walks with length 2

String Decimal Size Map

.00,.03,.06,.09,.30,.33,.36,.39, [0 0] 16 .60,.63,.66,.69, .90,.93,.96,.99

.01,.04,.07,.31,.34.37, [0 1] 12 _/ .61,.64,.67,.91,.94,.97

.02,.05,.08,.32,.35,.38, [0 2] 12 .62,.65,.68,.92,.95,.98

.10,.13,.16,.19,.40,.43, [1 0] 12 .46,.49,.70,.73,.76,.79

[1 1] .11,.14,.17,.41,.44,.47,.71,.74,.77 9

[1 2] .12,.15,.18,.42,.45,.48,.72,.75,.78 9

.20,.23,.26,.29,.50,.53, [2 0] 12 .56,.59,.80,.83,.86,.89

[2 1] .21,.24,.27,.51,.54,.57,.81,.84,.87 9

[2 2] .22,.25,.28,.52,.55,.58,.82,.85,.88 9

18 2.2 Substrings

Definition For a string x = [x1 x2 ... xa xa+1 ... xb−1 xb ... xn] the substring, x[a,b] from position a to position b of string x is

x[a,b] = [xa xa+1 ... xb−1 xb], where a, b ∈ N.

We shall say string x[a,b], instead of sub-string from [a, b] of x. If a = 1 and b = l(x), then this is the complete string. Sub-strings are used to look at intervals of strings with unbounded length. Substrings are helpful to analyze sub blocks of strings

Examples

1.

(π − bπc)[1,7] = [1 1 1 2 0 2 0]

Begin by identifying this as a string of the first 7 digits of π. Strings are

defined only for the interval (0, 1), hence

π − bπc = 3.14159265359... − 3 = .14159265359....

Since the string is the first 7 digits, the

d1 = 1, d2 = 4, d3 = 1, d4 = 5, d5 = 9, d6 = 2, d7 = 9,

thus the

δ1 = 1, δ2 = 1, δ3 = 1, δ4 = 2, δ5 = 0, δ6 = 2, δ7 = 0.

Meaning the string sub-string of the first 7 digits of π is [1 1 1 2 0 2 0].

19 2. Show the following

log10 2[1,12] = [0 0 1 0 2 0 0 0 2 0 0 1]

Notice log10 2 ∈ (0, 1). Since the string is the first 12 digits

d1 = 3, d2 = 0, d3 = 1, d4 = 0, d5 = 2, d6 = 9, d7 = 9,

d8 = 9, d9 = 5, d10 = 6, d11 = 6, d12 = 4, meaning the corresponding

δ1 = 0, δ2 = 0, δ3 = 1, δ4 = 0, δ5 = 2, δ6 = 0, δ7 = 0,

δ8 = 0, δ9 = 2, δ10 = 0, δ11 = 0, δ12 = 1.

Thus the string of the first 12 digits of log10 2 is

[0 0 1 0 2 0 0 0 2 0 0 1].

20 3. Show the following 1 = [1 0 0 1 0]. 27 [3,7] 1 First, 27 ∈ (0, 1). Now notice the string begins with the third digit from 1 27 = 0.037037037... and ends on the seventh digit, so

d1 = 7, d2 = 0, d3 = 3, d4 = 7, d5 = 0,

thus,

δ1 = 1, δ2 = 0, δ3 = 0, δ4 = 1, δ5 = 0.

So the sub-string of our repeating decimal is

[1 0 0 1 0].

2.0

1.5

1.0

0.5

1 2 3 4 5

Figure 2.1: The Rational Map of 1 27 [3,7]

On the next page there will be rational mappings of repeating rationals numbers and irrational numbers.

21 2.3 Some Walks

The number 1/7 = 0.142857142857... repeats itself every 6 digits, producing a cyclic type graph. The string our repeated decimal corresponds to is [1 1 2 2 2 1]. Notice how there is no drift on this graph that is because [1 1 2 2 2 1] has as many ones as it does twos.

2

1

5 10 15 20 25 30 -1

Figure 2.2: The Rational Map of 1 7 [1,30]

When plotting the first 100 realize there is no pattern. Since the irrational has unbounded length this is an interval of the number’s mantissa. This graph is shown in Figure 2.3.

10 8 6 4 2

-2 20 40 60 80 100 -4

√ Figure 2.3: The Rational Map of 2[1,100]

22 When looking at the walk of an irrational number it is helpful to look at a few of the maps, to see what is occurring the short term and long term. Notice when looking at the 100 step rational map of π there does not seem to be any pattern.

When looking at the longer maps, there does not seem to be a pattern but it does seem that the walk is hugging the axis. The strings of π are shown in Figures below.

5

20 40 60 80 100

-5

-10

Figure 2.4: The Rational Map of π[1,100]

30 20 10

-10 200 400 600 800 1000 -20 -30 -40

Figure 2.5: The Rational Map of π[1,1000]

23 2.4 String Analysis

We now consider some measures of displacements for walks of strings.

Definition The variation of string x[a,b] is n 2 X 2 V (x[a,b]) = ||x[ a,b] || = ((δi − δi ) mod 3) i=1

Notation For a string, x, then V (x[a,b]) = ν[a,b] the order of operations is to square, mod then sum in that order.

Examples Find the variation of the strings [1 1 1],[1 2 0 1], and [0 0 0 2 0 0 1 0 0 1].

2 ν[1,3] = ||[1 1 1] || = 3

2 ν[1,10] = ||[0 0 0 2 0 0 1 0 0 1] || = 3 The variation of a string will be, primarily, used with other operations.

Definition The amount of two’s in our string x[a,b] is defined by

n 2 X 2 τ[a,b] = ||x[a,b] − x[a,b]|| = δi − (δi mod 3). i=1

Definition The amount of one’s in our string x[a,b] is defined by

ι[a,b] = ν[a,b] − τ[a,b].

Definition The amount of zero’s in our string x[a,b] is defined by

ω[a,b] = l(x[a,b]) − (ι[a,b] + τ[a,b]).

Keep in mind ω[a,b], ι[a,b], τ[a,b] ∈ N0, is a count of the zero’s, one’s and two’s in a given string.

The triple (ω[a,b], ι[a,b]τ[a,b]) this is called the decomposition of x[a,b].

24 Examples

1. Find the decomposition of string a = [2 1 1 2 1 1 1 0 0 0]. First find,

2 τa = ||a − a ||

= ||[2 1 1 2 1 1 1 0 0 0] − [1 1 1 1 1 1 1 0 0 0]||

= ||[1 0 0 1 0 0 0 0 0 0]|| = 2

It is useful to use τa to obtain ιa,

ιa = νa − τa

= ||a2 || − 2

= ||[1 1 1 1 1 1 1 0 0 0]|| − 2

= 7 − 2 = 5

So,

ωa = l(a) − (τa + ιa)

= 10 − (2 + 5) = 3

Thus the string a has decomposition (3, 2, 5). The decomposition could be

used in the size equation,

s(a) = 35+243 = 139, 968,

thus there are 139, 968 decimals that are equivalent to this string.

25 2. Find the decomposition of string b.

b = [1 0 2 2 2 2 2 1 1 2]

First,

2 τb = ||b − b ||

= ||[1 0 2 2 2 2 2 1 1 2] − [1 0 1 1 1 1 1 1 1 1]||

= ||[0 0 1 1 1 1 1 0 0 1]|| = 6

so,

ιb = νb − τb

= ||b2|| − 6

= ||[1 0 1 1 1 1 1 1 1 1]|| − 6

= 9 − 6 = 3,

and,

ωb = l(b) − (τb + ιb)

= 10 − (6 + 3) = 1.

Thus our string b has a decomposition (1, 3, 6). Thus gives,

s(b) = 33+641 = 78, 732,

so there are 78, 732 decimals that are equivalent to this string.

Between string a and string b notice that s(a) > s(b) although l(a) = l(b).

26 2.5 Characterizations of Strings

The decomposition of a string gives a useful characterization of the drift of a walk.

There are characterizations of strings based on the decomposition of a given string.

The characterizations are used to better categorize strings.

Terminology If the decomposition of string x[a,b] has

ι[a,b] = τ[a,b],

then the string is a neutral string. An example of this is Figure 2.1.

Terminology If the decomposition of string x[a,b] has

ι[a,b] ≥ τ[a,b],

then the string is a positive string.

An example of this is Figure 2.2.

Terminology If the decomposition of string x[a,b] has

ι[a,b] ≤ τ[a,b],

then the string is a negative string.

An example of this is Figure 1.3.

In the examples from the previous section both strings a and b are both negative

strings.

Example

Characterize the string

C[1,48] = [12012012010111210111210111210202122202122202122]

First, the string must be decomposed,

2 τ[1,48] = ||C(10)[1,48] − C(10)[1,48]|| = 19

27 Then,

ι[1,48] = ν[1,48] − τ[1,48] = 19 so,

ω[1,48] = l(C(10)[1,48]) − (τ[1,48] + ι[1,48]) = 10

Giving a decomposition of (10, 19, 19), since

ι[1,48] = τ[1,48]

the string C(10)[1,48] is a neutral string. Since irrational walks are unbounded this helps determine if there is a drift in the walk. This is shown in figure 2.6

10

8

6

4

2

10 20 30 40

Figure 2.6: The Rational Map of Champernowne(10)[1,48]

28 Below there will be a definition of a not so common constant, then there will be plots of the sub-strings associated with the constant.

Definition For the continued fraction,

1 x = a0 + 1 a1 + 1 a2+ 1 a3+ ... it is almost everywhere, true that

1 lim (a1a2...an)n = K0 n→∞

where K0 is Khinchin’s constant.

∞ Y 1 K = (1 + )log2 r ≈ 2.6854520010 0 r(r + 2) r=1 A continued fraction is an expression obtained through an iterative process of

representing a number as the sum of its integer part and the reciprocal of another

number, then writing this other number as the sum of its integer part and another

recipricol and so on. Continued fractions that do not lead to Khinchin’s constant

are the rationals, and solutions to quadratic equations. It is not known if the

Khinchin constant is rational, irrational, or transcidental. The rational map of the

first 100 steps of Khinchin’s constant do not show any real pattern. When looking

at the first 1000 steps, notice there is still no real pattern. Notice the first 100

steps of Khinchin’s constant is a negative string. From the first 1000 steps, observe

that the walk, tends to the axis. The strings of Khinchin’s constant are shown in

Figures 2.7 and 2.8.

29 20 40 60 80 100

-5

-10

-15

Figure 2.7: The Rational Map of Khinchin’s constant first 100 decimals

30 20 10

200 400 600 800 1000 -10 -20 -30 -t40 •·~

Figure 2.8: The Rational Map of Khinchin’s constant first 1,000 decimals

30 Chapter 3

Analysis of Normal Strings

We will now consider string analysis and rational walks for an interesting set of irrational numbers.

3.1 Normal Numbers

This section details what is already known about normal numbers[6]. Acknowledge that Champernowne(10), is one of the few confirmed 10-normal numbers, besides √- the Copland- Erdos Constant. Where it is not currently known if 2 and other commonly used irrationals are 10-normal. Before defining normal and simply nor- mal by definition I would like to informally say that simply normal could be a terminating rational where as normal has to be non-repeating, thus irrational.

31 Let w be a fixed word of finite length m, and choose and fix integers n ≥ m, as

b well as a real number x ∈ [0, 1]. A count function, Nn(x; w) to be the number of

times the word w appears contiguously among (x1, ..., xn). Definition A number x is said to be simply normal in base b if

N b(x; j) 1 lim n = for all letters j ∈ {0, ..., b − 1}. n→∞ n b

That is, x is simply normal in base b when, and only when, all possible letters in the alphabet {0, ..., b − 1} are distributed equally in the b-ary representation of x.

Example The number .012345012345(base 6) is simply normal in base 6. Defi-

nition A number x is said to be normal in base b if given any finite word w with letters from the alphabet {0, ..., b − 1},

N b(x; w) 1 lim n = , n→∞ n b|w| where |w| denotes the length of the word m.

With this definition of normal the only proven normal-10 numbers are the

Champernowne(10) and Copland- Erdos constant.

Definition The Champernowne constant base b can be expressed exactly as

∞ X n C(b) = (P n log (k+1)) b k=1 b n=1 Examples

C(3) = .1201111221...

C(10) = .1234567891011121314...

Definition The Copland-Edros constant can be expressed exactly as

∞ P n X −(n+ log10 pk ) pn10 k=1 = 0.2357111317... n=1

32 3.2 Normal Strings

This section is looking at the rational walks of the Champernowne(10) and the

Copland-Erdos constant number to obtain insight on what 10-normal strings look like. From observation notice how the walks of Champernowne(10) are self similar, likewise with the Copland-Erdos constant.

Observe that Champernowne(10) is naturally made of neutral sub-strings, that is why the walk keeps returning to the axis.

120 100 80 60 40 20 -20 200 400 600 800 1000

Figure 3.1: The First 1,000 Steps of Champernowne(10)

1200

1000

800

600

400

200

2000 4000 6000 8000 10 000

-200

Figure 3.2: The First 100,000 Steps of Champernowne(10)

33 The rational mapping of 1, 000, 4, 000, 8, 000 and 25, 000 are provided. Notice that each mapping looks like the next. Since the number was proven to be 10- normal, it should begin to decrease in height at one point, but due to computation time there was no way to produce an image of this.

300

250

200

150

100

50

200 400 600 800 1000

Figure 3.3: The First 1000 Steps of the Copland-Erdos Constant

600 500 400 300 200 100

1000 2000 3000 4000

Figure 3.4: The First 4000 Steps of the Copland-Erdos Constant

34 1500

1000

500

2000 4000 6000 8000

Figure 3.5: The First 8000 Steps of the Copland-Erdos Constant

4000 3000 2000 1000

5000 10 000 15 000 20 000 25 000

Figure 3.6: The First 25000 Steps of the Copland-Erdos Constant

35 Bibliography

[1] Leslie Lamport, LATEX: A Document Preparation System, Addison-Wesley Pro- fessional, 2nd Edition, 1994

[2] D. G. Champernowne, The construction of decimals normal in the scale of ten,

Journal of the Londen Mathimatical Society, Volume s1-8, Issue 4 (1933), page

254-260.

[3] Kaurow, Vitality. Visualizing Digits of Pi with Colored Walks. Visualizing

Digits of Pi with Colored Walks - Online Technical Discussion Groups-Wolfram

Community, community.wolfram.com/groups/-/m/t/82377.

[4] rtacho, Fancisco, et al. Walking on Real Numbers. Davidhbailey.com, 23 July

2012, www.davidhbailey.com/dhbpapers/tools-walk.pdf.

[5] Weisstein, Eric W. ”Copeland-Erds Constant.” From MathWorld–

A Wolfram Web Resource. http://mathworld.wolfram.com/Copeland-

ErdosConstant.html

[6] Khoshnevisan, Davar. Normal Numbers Are Nor-

mal. CMI Annual Report, 2006, pp. 1531.,

http://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf

36 Appendix A

Code

steps = 2; number = .27;

rules := {0 -> {1, 0}, 1 -> {1, 1}, 2 -> {1, -1}}~Join~(Rule @@@

Thread[{Range[4, 7], Tuples[{1, -1}, 2]}]);

onEsteP[pt_, st_] := pt + (st /. rules); ptsl := FoldList[onEsteP, {0, 0}, Mod[RealDigits[FractionalPart[number]

, 10, steps][[1]], 3] ];

Graphics[MapIndexed[{Opacity[1],

ColorData[21][#2[[1]]/Length[ptsl]], Line[#1]} &,

Partition[ptsl, 2, 1]], Background -> White,Axes->False]

37