A general Cantor set is a closed set consisting entirely of boundary points. Such sets are uncountable and may have 0 or positive Lebesgue measure. The Cantor set is the only totally disconnected, perfect, compact metric space up to a homeomorphism Willard Portions of this entry contributed by Margherita Barile. Boas, R. A Primer of Real Functions.
Washington, DC: Amer. Cullen, H. Introduction to General Topology. Boston, MA: Heath, pp. Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Harris, J. New York: Springer-Verlag, p. Sloane, N. Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, pp. Willard, S. So I decided that I would be the person to put this book together. However, in writing this manuscript for advanced undergraduates I left out many proof because they were too long and technical for this intended audience.
It is my hope that anyone interested in the details will be able to use the bibliography to go to the source and find out the nitty-gritty. As noted in the Preface, without my student Andy Brown and his interest in an independent study on Cantor Sets, I would not have thought of putting together my notes which then became this manuscript. Gary Grabner was a huge help with the figures. My biggest cheerleader and fan was my wife, Jacqueline Jensen-Vallin.
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That said, any and all mistakes are mine and mine alone. If you find any errors, or if I have missed your favorite topic, please don't hesitate to contact me. If it were, there would be much less to say. As an object itself, the set can be fascinating, but what is most appealing is not the set per se, but where the set can lead one. In analysis one uses the notions of measure, category, and cardinality to describe many objects.
When teaching those topics the Cantor Set always pops up as an example to fortify understanding of the concept. The text makes for a good ancillary in several different courses: analysis using Chapters 2, 3, 9 as a start , topology Chapters 2, 8, 9 , fractals Chapters 6, 7, 10 , and algebra Chapters 2, 5, 4.
This is a Senior-Level class that, while having a unifying theme, draws in material from the previous coursework for their undergraduates. These courses can include analysis, algebra, topology, and number theory. Several Capstone Courses also require an out-of-class project. This book has a unifying theme, of course, and shows how the Cantor Set appears in so many different mathematical topics, thus bringing together the collection of courses that make up the lion's share of the upperlevel math curriculum.
Given the wide range of topics covered, there are plenty of places for students to find topics and questions to study outside the classroom for the project portion of the course. This manuscript, which arose from an independent study course, is an example of the type of book needed for an independent study. There are so many ideas covered that whether a student has a preference for analysis or algebra or something else, that subject can be found. Lastly, this book is a resource for professors. As we all know, the "sage on the stage" reading from well-worn notes or overhead slides is an outdated idea.
Parsed into its separate chapters and sections, this book can be used to add to several different courses in the undergraduate mathematics curriculum. With the topics covered and detailed bibliography, one can find a myriad of directions to take one's research. By Robert W. Born 3 March in St. Petersburg, Russia, he was the oldest of six children.
Due to his father's ill health, the family moved to Frankfurt, Germany, hoping to find milder winters. In he started his higher schooling at the Federal Polytechnic Institute in Zurich. After the death of his father, Cantor moved on to study at the University of Berlin. Among his professors there were Kronecker, Weierstrass, and Kummer. This could not happen as Kronecker, who led the department at Berlin, was not supportive of Cantor and his work.
Kronecker went so far as to try and pressure Heine to not publish Cantor's paper "Uber trigonometrische Reihen" in the journal Mathematische Annalen. In a letter to Hermite, Cantor complained that Kronecker, among, other things, called his work "humbug. He carried on correspondence with some of the greater mathematicians of the day e.
He was very sensitive to even well-intentioned criticisms of his work. In , Cantor suffered his first attack of depression. Instead, Cantor believed they were written by Francis Bacon. Cantor was hospitalized again for depression in and then several more times starting in These sapped him of much of his zeal for mathematics.
However, his work continued to be appreciated. In he was awarded an honorary doctorate from the University of St. Andrews in Scotland. He officially retired in and died on 6 January in a sanatorium where he had been for the last year of his life. Cantor's contributions to mathematics are vast, with results in number theory his thesis topic and set theory which he founded.
Working on the infinite drew the ire of some mathematicians, philosophers, and religious scholars. For example, if a and b are two positive numbers, then a 3 Poincare said that in the future people would look on Cantor's work as "a disease from which one has recovered. However, Cantor was not without his supporters. At the Second International Congress held at the Paris World Exposition of , David Hilbert presented a list of the major unsolved problems of the time, hoping to spur interest in what he believed to be the most important problems of the day.
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The item at the top of the list was Cantor's Continuum Hypothesis. Hilbert is famously quoted as saying, "No one will drive us from the paradise which Cantor created for us. That means much of what we look at here the reader is familiar with. Thus there will not be a lot of proofs given. Most of these concepts are familiar, even if a bit of dust must be shaken off. Hopefully those ideas which are unfamiliar will be few and easily digested by the reader. A set is a well-defined collection of objects. We will use capital letters to denote a set.
As the name suggests, List Notation means the writer lists some of the objects in the set. Note that sets are written using braces for enclosure. This is a useful way of doing things as some sets defy listing. The colon is usually translated as "such that" or "so that," and then followed by a description. This is read as, "C is the set of x such that x is a letter in the English Alphabet. An object which is a member of a set is called an element of the set and usually written with a lowercase letter. The symbol for is an "element" of is s.
Rather than write, "a is a member of the set X," we just write "a G X. Subsets are used to prove equality of two sets. Given two sets A and B, there are several ways to create new sets. A and Z? A specific set difference which we will encounter many times is the complement of A. To learn more on set theory, we suggest [21 ] or . We shall look at some specific sets in the real line. Intuitively, a set A in R is bounded if there are numbers m and M where no element of A is below m or above M.
Then A is open but not an open interval , B is closed, C is neither open nor closed. Our first theorem, relates Ta and Qs sets and their complements. Theorem 2. Now each Of is a closed set, thus Ac is Fa. The set. A function which is both 1 — 1 and onto is called a bijection. We will use bijections when we discuss cardinality. A function does not have to be a bijection or have an inverse in order to talk about inverse images.
These restrictions can have important consequences. One special type of function we will use often is called a characteristic function. Some functions have a special property involving the distance between two inputs versus the distance between their respective outputs. Scratchwork: This is a little more difficult. Since the function is not linear, we cannot use straight factoring. However, factoring still plays a role here. Different spaces may have different idea of distance. These distance functions are called metrics. Every set can have a distance function defined on it as there is always the so-called trivial metric.
Although these definitions and concepts in this chapter are written for the real line, we will explore various generalizations. Give an example of a non-empty, open set in R whose boundary is the empty 2. Well, we hope to find answers to that question as we continue to read this book. One thing we can say is that the set seems to pop up everywhere. When looking for an example to illustrate a definition some kind of Cantor Set usually does the trick. This can happen in real analysis, topology, abstract algebra, probability theory, fractal geometry, the list go on.
In searching for extreme examples, many times one begins, "Start with a Cantor Set and For the most part, we will drop this word and leave it to the reader to understand when we mean this set and when we mean some general Cantor Set. Divide it into three equal sections and remove the open middle third.
We then continue inductively. At step n, Kn consists of T1 closed subintervals. Is there something magical about dividing things into thirds?
No, of course not. One way to generalize this is to change the ratio of the interval removed. Normally we write numbers in base With base 10 we use the digits 0,1, 2, 3 ,. While the symbol remains the same, the meaning changes with its position and is interpreted as a multiple of a power of Similarly, a number to the right of a decimal point refers to negative powers of ten; for the number 0. If we let b be an integer greater than 1, we can write numbers in base b, using powers ofb rather than 10 and b digits 0 through b.
In base 5 the digits are 0, 1, 2, 3, 4; In base 2 binary the digits are 0, 1; in base 12 duodecimal we use, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, and E. We will use the subscript b to say a number is written in that base. So s refers to base 5. Converting from base 10 into another base can be done by finding the most copies ofbn that "fit" into the numberfor each positive integer power ofn.
A second way to do this is with a division trick, looking at the remainders from repeated division by 5 in this case since we 're converting into base 5. We stop here since the dividend is 0. To convert a base 10 decimal into another base, b, we multiply repeatedly by b and collect whole number parts. This means the first spot to the right of the base 4 quaternary point is a zero of course we cannot say the "tenths place" as we are not in base Then 0.
After removing the 3 there is 0. This method uses factoring. The construction of a Cantor Set fits right into base 3. When we divide the unit interval into thirds, we can look at this as splitting the numbers up according to their first digits. The left section consists of numbers whose first decimal digit in base 3 is a zero, the middle section has all numbers whose first decimal is 1, and the right section has numbers whose first digit is 2.
Removing the open middle rids us of all numbers whose first digit is a 1. When the left interval is subdivided into thirds and the middle removed, what we are keeping is all the numbers between 0 and 1 which begin in base 3 as. Technically, we have to be a little careful as some numbers are not unique in their representation. For example, in base 10 1. Now if we write our numbers between 0 and 1 in base 3, we can say the Cantor Set consists of all numbers which have a base 3 representation consisting only of 0 's and 2 's to the right of the ternary point and a 0 to the left of the ternary point.
Both of those are endpoints of intervals in some Kn. For an example of a non-endpoint in base 3 we have 0. Let us say a bit more about what goes on with points in the Cantor Set. This comes from  where the rational values in C are partitioned. These correspond to the left endpoints of the removed intervals in the Cantor Set construction. For example, 0. We can find this number's value in base 10 by relating it back to a purely periodic number.
Here we see 0. Is it a big set or small? Before we can answer this question, we must define what we mean by saying a set is small. This notion comes from Van Rooij and Schikhoff . We say a collection of sets S from K is a collection of small sets if 1. Some of the collections we shall look at that qualify as a small set include countable sets, measure zero sets, and first category sets. It is worth noting that finite sets do not meet this idea of a small set. As we shall see, in some ways the Cantor Set is small and in other ways large. Cantor's work in this area opened mathematics up to the ideas of what it means for a set to be infinite and the notion of different types of infinity.
First we need to recall two properties a function from Chapter 2, followed by examples. Definition 3. Then f is one-to-one. Then f is onto, but not one-to-one. Then f is a bijection. As we will see, cardinality, even if not known under that name, is something with which the reader is quite familiar. We say A is equivalent to B if there exists a bijection, f, between A and B. Let us take a step back and define equivalence relations and partitions before we move on. We write to say, "x is related to y. We say the relation is 1. These collections are called equivalence classes.
For two cards x and y, we say x is related to y if they are both diamonds or both spaces, both hearts, both clubs. This is an equivalence relation and the equivalence classes are the suits clubs, spades, hearts, and diamonds. If instead x is related to y by them having the same value, then there would be 13 equivalence classes: aces, deuces, treys, We say p x is related to q x if they have the same y-intercept.
This is an equivalence relation and the equivalence classes are the functions which all agree at 0. I The theorem below gives us three important properties for equivalence classes. Casually, these say equivalence classes are like the pieces of a jigsaw puzzle. Each piece contains part of the picture, no two pieces contain the same part, and together they make up the whole scene. Theorem 3. Proof: We will prove part 1 only.
Parts 2 and 3 will be left for homework. Thus the symmetric property is satisfied. Finally, if f is a bijection between A and B and g is a bijection between B and C, then the composition go f is a bijection between A and C. This means we have equivalence classes and that is where we willfindcardinality. Otherwise A is an infinite set. When the latter is true the set is called countably infinite. If a set is not countable, then it is called uncountable. For countably infinite we use Ho. The collection of countable sets are our first type of small set using the criteria of .
Let us look now at examples of countable and uncountable sets and the proofs of their various cardinalities. But before our first result, let us state a lemma. Its proof is left as a homework exercise, but we shall use it right away in Theorem 3. Lemma 3.
Then a diagonal argument puts these element m l — 1 correspondence with N. There is nothing special about having only two sets, so this can be fully from an ordered pair to an n—tuple by the following. Proof: This is yet another example of a diagonal argument and will be left as an exercise. In Cantor published a proof that an interval of the form [a, b] is uncountable. Later, in , he published another paper containing his famous "diagonalization" proof of this.
We give our own version of the proof now. Proof: Every number in the unit interval can be written as 0. This is the only possible redundancy in representations. Proceeding with a proof by contradiction, suppose the numbers in [0,1] are countable. This contradicts our list being a complete roster of the numbers in [0,1]. Thus the unit interval is uncountable. Proof: We do this by contradiction. Similarly a subset of a countable set must be countable.
Corollary 3. We say x is a algebraic number if it is the root of a polynomial with integer coefficients. If a number is not algebraic, then it is transcendental. Well-known transcendental numbers include ir and e. There are many, many others. The proofs for n and e being transcendental are non-trivial and will not be presented here. Proof: Each algebraic number a is associated with a polynomial p x with integer coefficients. So we will show this collection of polynomials is countable. Finally this brings us that the set of algebraic numbers is countable. Proof: This is the same as the proof for the irrational numbers.
This really shows the paradoxical nature of the set. We start with the unit interval, [0,1], remove open intervals themselves uncountable whose total length is one, yet what is leftover has the exact same cardinality as the unit interval. Proof: As we have already seen, the Cantor Set can be represented by the set of all numbers between 0 and 1 which have a base 3 representation that consists of only 0's and 2's. An important application of countability follows in Theorem 3. It is used in real analysis to help describe what we refer to as "exceptional sets.
There are more infinite cardinal numbers than the ones we have looked at. In fact, there is an infinite hierarchy. This was the first of 23 problems that David Hilbertpresented at the World's Exposition as the most important mathematics problems for the upcoming 20th century. Thus we have a statement that it is impossible to prove or disprove. A countable set is in some way small, while an uncountable set is large. Another way of small versus large is the idea of category. Category is part of the larger study of topology.
In topology we generalize the idea of open sets. Here is the modern definition. The sets in T are called the open sets in S and T is called a topology on S. When we refer to a space as a topological space, it is a pair S,T. So we get to define what it means for a set to be open. This is called the trivial also the indiscrete topology on S. This is called the discrete on S. Let us look at how this generalizes what we know about K. Formally we say the following: Definition 3.
We say xn converges to. Open intervals are what we generalized to get to open sets in a topological space. This leads us to this definition of convergence. As the examples below show us, strange things can happen depending on the topology on the space. Before the first example we need a definition. Thus the only convergent sequences are the eventually constant ones.
Although the notation is the same, this is not the inverse function. In addition, we say f is a continuous function if it is continuous at each point in its domain. The example below shows how this works and some of the strangeness involved. If things reverse themselves and the domain is V and range S, then not only is f continuous, but every function is continuous.
To get the full power of our big result The Baire Category Theorem we need the additional property of a space being complete. We are saving that particular property for the chapter on Self-Similar Sets. So in a bit we will switch to only working in M with the usual, Euclidean, topology. That is, every open set contains at least one point in A.
Now we define an idea for a sparse type of set. We are trying to make things so that no part of the set can be thought of as dense. A set can be neither dense nor nowhere dense. It is an exercise in the homework to come up with an example of such a set. This idea is not small in the Van Rooij and Schikhoff way. It is not true that the countable union of nowhere dense sets will be nowhere dense. This is a stepping stone to getting such a smallness.
If a set is not first category, then it is called second category. Now let us concentrate on just R with the Euclidean topology. Any finite set will be nowhere dense. The integers are an example of a set which is infinite and also nowhere dense.
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The irrational numbers are dense in the line as are the rationals. The notions of dense and nowhere dense are not "opposite " in the strictest sense. Finally, sometimes sets that are not nowhere dense can be broken down partitioned into pieces which are nowhere dense. The canonical example of a first category set which is dense in the line is the rational numbers. However, every non-empty, open interval in the line contains rational points. For the most famous application of category, we need to delve a little more deeply.
There are other ways to state this. Alternatively, the complement of UAk must be a nowhere dense set. We will use this idea to prove a result at the end of this chapter. In alternating moves the players choose closed and nested intervals. We then look at the point x e nifc. The question is, "Under what circumstances is Player I guaranteed to have a winning strategy?
Thus some proofs that a set is second category are proofs that Player I can win the Banach-Mazur Game. The origin of this result is worth looking into as a fascinating glimpse into the culture of mathematics. The idea of the game was first posed in the Scottish Book. From to , a group of mathematicians from the University of Lwow in Poland at the time, now part of Ukraine would gather at The Scottish Cafe House.
They would discuss their various works and, if an interesting problem arose, would signal the waiter to bring out a notebook kept behind the bar. Problems would be written in the notebook. Some of the problems had prizes attached to them. For more on this fascinating time see . Problem 43 contained the beginning of the game and proposed by Mazur.
It was solved by Banach with the prize being a bottle of wine , but he never published his proof. Let us turn our attention back to the Cantor Set. Such a proof using this theorem can be done implicitly, that is, without actually having to construct such a function. This is not exactly part of the theme of this book, so we will leave it alone, but it is a very interesting result which should be a part of any course in real analysis.
A second application is an interesting property of continuity. We begin with Thomae's Function also called the Popcorn Function as well as other names. This function is continuous exactly on the irrational numbers. For irrational x, any rational x nearby must have increasingly larger denominators showing f is continuous on the irrationals.
Interestingly enough, we cannot reverse the roles of sets on which a function is continuous. The proof involves the Baire Category Theorem. This lemma states that the rational numbers as a subset ofM. Recall a set is Qs if it can be written as the intersection of a countable collection of open sets. Proof of Lemma We will do this as a proof by contradiction. However, for V the intersection is empty one grouping has exactly the rational numbers, the other misses the rationals, one at a time.
Thus we have proven our result. This is a special type offunction between two sets A and B. An interesting result involving homeomorphism is a relationship between Cantor Sets and compact metric spaces. Metric spaces were lightly introduced in Chapter 2. Compact sets are not yet defined, but in the real line any closed and bounded set is compact. We state this theorem without proof, but will revisit it, with proof, in Chapter 6 after more is said about metric spaces.
As for the theorem: Theorem 3. This is proved later in Chapter 6. There are various ways to come up with measures for a set e. For now, we will concentrate on Lebesgue measure, which Lebesgue developed in for his PhD thesis "Integrate, longueur, aire Integral, Length, Area ". We begin with the basics. This also works for open or half-open intervals. We would like to generalize this to come up with an idea of "length "for any set. Instead of length, we call it measure and denote measure of a set E by m E. The nice properties for measure to ideally have are: 1. Measure is defined for each set E of real numbers.
Unfortunately, it turns out to be impossible to construct a function on sets having all four of these properties. Something must be sacrificed. Before we can talk about measure we need to define an intermediate property called the outer measure. It is trivial to show that both 0 and K are both measurable sets. Note here that the idea behind a proof of measurability of a set usually consists of two pieces. However, part of this we get for free. Thus we only need to show the other direction. To see this in action, we shall prove that any set with outer measure zero is a measurable set. Then E is, in fact, a measurable set.
Proof: Let A be an arbitrary set. Since A was arbitrary, this works for all A and we have proven E is measurable. That the usual Cantor Set C is measurable is not a shock. It is possible to modify the Cantor Set construction so as to get a Cantor Set with positive measure. One construction of such a set, called a "Fat" Cantor Set, is in the exercises and Chapter We will derive an example to show that Property 1 does not hold; that is, exhibit a nonmeasurable set.
This is a constructive development, but not explicitly so. Before we begin, though, recall the definitions for equivalence relations and partitions. For any two numbers x and y in the unit — x is a rational number. This is an equivalence interval we say x is related toyify relation reflexive, symmetric, and transitive see the exercises which means we can partition the unit interval into classes, collections of equivalent numbers. Our claim is that these Sr are nonmeasurable.
We show this in the theorem below. Proof: We will prove the result for E U F and leave the others as exercises. Note that these results tell us that measurable sets form a a-algebra. Then both UnEn and nnEn are measurable sets. Let us end this part of the discussion with an application of measurable sets. As we know from calculus, if a function f is nice, we can find its Riemann integral. In the beginning we have continuous and positive functions for "nice" and introduce this integral to solve The Area Problem.
Then our ideas are generalized so that functions neither need to be continuous nor positive-valued. This is presented without proof. In order to get there, we shall now move on to the idea of measurable functions. Characteristic functions are a big part of Lebesgue integration. An unproved by us corollary of this is that any continuous function is measurable.
This is a consequence of the following theorem. Then the following are equivalent: 1. Proof: We will show the first implication. The rest comes from complements. If f is Riemann integrable on I, we denote R j f as the Riemann integral of f. This integral is called Darboux integral. Since the functions integrable using Darboux's definition and functions integrable using Riemann's definition are the same, we will again use R f notation. It is a fairly straighforward Advanced Calculus exercise to prove that the Riemann and the Darboux integrals are the same. Although the Riemann definition is typically what one sees in a Calculus course, the example below will show us the usefulness of the Darboux definition.
Notice that there are some restrictions in these definitions. The function f has to be bounded. These two conditions, while necessary, are not sufficient to make a function integrable. These are basically sums of characteristic functions. This is called the canonical representation of g. There are other ways to represent g. Although this seems unusual, we will have need of this in proving theorems.
We see this in the next example. Consider A to be the integers.
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Then this simple function is not a step function. Before we prove it, though, we need this lemma. Proof: This proof is left as a homework exercise. We are now ready to define a Lebesgue integral for a bounded function f. We do this by approximating with simple functions similar to the way step functions are used to approximate the Riemann integral. This is an extension of the Riemann integral. If f is Riemann integrable on [a, b], then it is measurable and! So have not yet arrived at the finished Lebesgue integral. The extras that we must dispose of are the requirements that m E is finite and that f is bounded.
We can rid ourselves of both, but at the cost of a new condition. On the other hand, f is constant at 1 except on a set with measure zero. These notions are mostly unrelated to one another: A set could be small in one way category yet large in another cardinality. We wish to end this chapter with a striking result that can be found in Oxtoby , taking the entire space we are working in and decomposing it into two small pieces.
Each Oj is an open set since it is the union of countably many open intervals and Q C Oj. What about 0? Hint: This also has to do with a base other than This time it is base 2. Thus a set can be small in one way category while large in another countability. There are also sets which are both open AND closed. They are called "clopen. The survivors spend the day gathering a pile of coconuts. They will divide the coconuts up the following day.
During the night one man woke up and started worrying about the coconuts. He decided to take his share, and divided the coconuts equally into five piles. There was one left over which he gave to the monkey, then he took his share and hid them away, before going back to sleep. A little later a woman woke up and had the same fear. She too divided the coconuts into five piles, and had one left over that she gave to the monkey, before hiding her share and going back to sleep. And so the night went on, with each person waking up, dividing the remaining coconuts into five piles, hiding one share, and giving the one extra to the monkey.
In the morning, they awoke and divided the remaining coconuts, which came out tofiveequal shares. How many coconuts were there to start with? A continued fraction is similar to that idea of fractions within fractions. Definition 4. Read these books and papers carefully. This requires just an application the Division Algorithm. First, for a rational number in lowest terms this process is guaranteed to terminate.
Notice that in our dividing process the remainders are strictly decreasing. This means they must at some point become the number 1. So although we are not writing this as a formal theorem, every rational number can be written as a finite simple continued fraction. We shall then require in our representation [a0 : ai, a 2 ,.
It turns out that these very intriguingly partition into different types. The value TT, the all-purpose number, is an example of an irrational number that is not a quadratic irrational. Let x be a positive irrational number. Lagrange proved in that the continued fraction expansion of any quadratic irrational will eventually become periodic. It is also worth noting here a way in which continued fractions and rational numbers differ.
If the decimal expansion of a number is periodic, then the number is rational. How about going the other direction? When we have an ordinary real number but with a repeating decimal representation, we are able to find its rational number representation, can we do the same with a repeating continued fraction? The answer is yes. This does not help us solve for x.
To discuss continued fractions a little more, we need to add to our terminology: We need to bring convergents to the mix. Typically, we represent these convergents as the rational numbers that they are. The relationship between the values of the p's, q 's, and a's is an inductive one, summarized in the theorem below. Theorem 4. Now we have a continued fraction with only k terms. Proof: This is left to the reader. It is another induction proof. A consequence of Theorem 4. A wellknown theorem from real analysis e. We know that for rational x the simple continued fraction expansion we get from x is equal to x when simplified.
The theorem below explains that the sequence converges to x. In general, we can think of a Cantor construction by starting with a closed interval [a, b] and removing an open middle portion of the interval, thus creating two closed subintervals. Continuing this process leads us to a Cantor Set. The point of this section is to show that we can make such a construction utilizing continued fractions. There are two types of intervals with rational endpoints that we need to consider when looking at what is in S k.
Once we have a closed interval, when we remove the middle subinterval, of the type Form I we remove the interval [0 : a i , a 2 , a 3 ,. In both instances, the removal of the open interval leaves an interval of the first form on the left side and an interval of the secondform on the right. Thus we can continue the process, and the resulting set consists of the numbers described by S k. Notice here ao can have any value. If ao is fixed at n, then the resulting set is referred to as F n, N. So let us take a quick glimpse into these, starting with an example and a definition.
This integral solution problem is the heart of the study of diophantine equations. The name honors Diophantus, a Greek mathematician who wrote about these equations in a series of books called the Arithmetica. Because x has the smaller coefficient. So letting u run over all possible integer values, we create a family of solutions to this equation. The Euclidean Algorithm is also what we use when we find the integers we need for the continued fraction representation of a number.
Therefore, we cannot be too surprised that we can talk about the solution to these equations using continued fractions.
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This equation is guaranteed to have infinitely many solutions by the Euclidean Algorithm. From Theorem 4. This depended on n being even. Now we have to turn our particular solution into a general solution. This is where gcd a, b — 1 comes into play. The right side is a multiple ofb, but no factor ofb can be a factor of a. The first one joins together continued fractions, and self-similar sets from Chapter 6 and comes from . The proof of this result can be found in that book. Then F is a fractal with 0. Let us apply this theorem in an example.
The theorem will enable us to find a beautiful representation for a nontrivial number. If there is no solution, explain why. Find the first three convergents in the simple continued fraction expansion of 4. What pattern Use Theorem 4. Show that F in Example 4. Our study of p-adic numbers will merge from two different directions.
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On one hand, it can be looked at as part of abstract algebra, and for that we will review some basics in the topic; on the other hand, p-adic analysis is a generalization of "normal" Euclidean analysis on the real line. And, of course, in the end either direction brings us around to the Cantor Set. Abstract algebra sometimes called modern algebra arose from many sources: the study of linear equations, the rise of non-Euclidean geometry, and work on solving Diophantine equations, to name a few.
Our main concern here is to introduce groups and then move on to rings and fields. Many books refer to the operation as multiplication, although this does not necessarily mean multiplication of numbers The Elements of Cantor Sets -With Applications, First Edition. Definition 5. The abstract notion of groups first appeared in papers by Arthur Cayley in Let C be the set of all bijections 1 — 1 and onto functions with domain and range M. In fact, this is an example of a finite group since it has a finite number of elements. In the examples above, all groups but C, o have the commutative property.
If a group has this property, it is called an abelian group after Neils Henrik Abel. If a group is not abelian, it is called non-abelian. A simple introductory theorem is as follows: Theorem 5. There is only one identity element. For each a G G, the inverse of a is unique.
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Proof: We will prove the first and fourth parts, leaving the remaining two as an exercise. Suppose G has two identity elements, e and e. The fourth part has two parts to the proof. We get that from the definitions of what makes a group. As there is more structure to a ring, there are also more requirements. If the ring does have such an element, then we say R is a ring with unity.
Additionally, the multiplication does not have to commute. Here are two more properties, important enough to get their own definitions. The integers are an integral domain. Some simple results about rings are in the theorem below. The proofs are left as an exercise. Theorem 5. Then 1. The integers are not a division ring. An example of a ring that is not an integral domain are the set of 2 x 2 matrices with matrix addition and multiplication.
The rational numbers, Q, with the usual operations of addition and multiplication are a division ring. Finally we come to what we really need, namely, fields. The rational numbers, real numbers, and complex numbers are all examples of fields. We start with the integers Z. Given two integers a and b, we say a divides b if there is an integer k such that b — ak. Thus 4 12 and 6 42 while 5 j Notice that this definition uses multiplication and not division.
The collection of equivalence classes is a finite set as each integer ends up being congruent to its remainder on division by 4. So the only equivalence classes are , , , and . Do we then have afield? It depends on n. However, if n is a prime number usually primes are denoted by p we do get afield. This equivalence relation creates equivalence classes and M. This section develops another idea of distance, creating a new way of looking at rational numbers and their completion. In the next section we will then relate the p-adic integers to Cantor Sets.
We start by fixing a prime number p. In algebra, a valuation is a function that measures the size of elements in afield. We define norms right now. We now take this norm and turn it into a metric on the rational numbers, the same way we would with the ordinary absolute value. Thus we have the metric space as Q, dp. Two numbers are called relatively prime if their greatest common divisor is one. Let us look at the third. Ultra-metric spaces have many interesting properties, but we will leave you with this one to ponder: In an ultra-metric space, all triangles are isosceles.
The position we are now in is a familiar one from a course in real analysis. One way to fix this is to relate Cauchy sequences. This leads us to the completion of the rational numbers, which we call the real numbers, R. As alway, start with a fixed prime number, p. We will denote this set by Q p. More about rational and irrational numbers in Q p will be in Section 5. We will see how they differ from our usual concept of integer and, in the end, are a non-trivial example of an abelian group.
Recalling infinite series, what we are dealing with here is the limit of a sequence of partial sums.