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Continued Fractions
In this elementary-level text, eminent Soviet mathematician A. Ya. Khinchin offers a superb introduction to the positive-integral elements of the theory of continued functions, a special algorithm that is one of the most important tools in analysis, probability theory, mechanics, and, especially, number theory.
Presented in a clear, straightforward manner, the book comprises three major chapters: the properties of the apparatus, the representation of numbers by continued fractions and the measure theory of continued fractions. The last chapter is somewhat more advanced and deals with the metric, or probability, theory of continued fractions, an important field developed almost entirely by Soviet mathematicians, including Khinchin.
The present volume reprints an English translation of the third Russian edition published in 1961. It is not only an excellent introduction to the study of continued fractions, but a stimulating consideration of the profound and interesting problems of the measure theory of numbers.
Presented in a clear, straightforward manner, the book comprises three major chapters: the properties of the apparatus, the representation of numbers by continued fractions and the measure theory of continued fractions. The last chapter is somewhat more advanced and deals with the metric, or probability, theory of continued fractions, an important field developed almost entirely by Soviet mathematicians, including Khinchin.
The present volume reprints an English translation of the third Russian edition published in 1961. It is not only an excellent introduction to the study of continued fractions, but a stimulating consideration of the profound and interesting problems of the measure theory of numbers.
112 pages, Paperback
First published May 14, 1997
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About the author
Aleksandr Yakovlevich Khinchin
10 books2 followersАлекса́ндр Я́ковлевич Хи́нчин
A.I. Khinchin
A.Y. Khinchin
A. Ya. Khinchin
A.I. Khinchin
A.Y. Khinchin
A. Ya. Khinchin
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Displaying 1 - 4 of 4 reviews
May 12, 2018
"I feel that an elementary monograph on the theory of continued fractions is necessary [...]" (p.ix)
Overview
Continued fractions (c.f.) can be used to represent real numbers. This well-written, 94-page book by Khinchin covers the basic facts about this correspondence as well as some applications in diophantine approximation and measure-theoretic questions about c.f.. The book was published first in 1935, the later editions (1949, 1961) without significant changes. Though probably many things have been discovered since then, the text provides a good entry to this field of study.
Content
There are three chapters. The first one gives some basic and very useful (in-)equalities.
The second chapter starts out with proving, that every real number x has a (under a reasonable convention) unique c.f. converging to it; the rationals being exactly those whose
c.f. is finite. Later in this chapter we see also a proof of Lagrange's theorem: The quadratic irrationals are exactly those with a periodic c.f.
The main topic though are inequalities of form |x-p/q| < f(q), where x is a real, p,q integers and f a positive function; here one asks for solutions in p, q. This formula occurs for example if one wants to estimate how well x is approximated by an initial segment of the c.f. representing x. It is proved that the best one can get is |x-p_k/q_k| < 1/q_k^2; p_k/q_k being the value of the initial segment of length k of the c.f. representing x.
Rearranging this inequality one enters the field of diophantine approximation. Since c.f. are in some well-defined sense optimal in approximating reals, they are a good tool to study such problems. We see a theorem of Chebyshev's on the number of solutions in p,q\in Z for |xq-p-a|<3/q (a arbitrary). A non-pigeonhole-principle proof of Dirichlet's theorem is given, based on an inequality of chapter one.
The third chapter is concerned with the size of sets of reals defined by some property of their c.f. expansion. For example, a criterion on f is given such that the set of reals x with infinitely many solutions of |x-p/q| < f(q)/q in p,q\in Z is the complement of a null set. Another big result is Kuzmin's theorem. While this itself is a rather technical statement, it can be used to answer questions like: What roughly is the measure of the set of reals in the interval [0,1] whose c.f. has a 5 at the 2018th place? And with some further work: how often on average does a given number n occur in the c.f. of a given real? In fact, it is proved, that outside a set of measure zero, this density does only depend on n, but not on the real! There is no proof of the existence/value of Lévy's constant (nor Khinchin's constant), but its existence is mentioned and references are given.
Style
The text (the translation) is a pleasure to read. The general direction of enquiry and the significance of particular theorems is discussed. Every once in a while there is a remark about the history of the problem at hand. There are no exercises. There is a one-page index, but no bibliography. But the text is short, so neither is a real issue. The book is elementary: the only "tools" needed are sequences, series and some common integrals. For chapter three you will need the definition and basic properties of the Lebesgue-measure. The proofs in that chapter are also considerably longer and more technical.
Overview
Continued fractions (c.f.) can be used to represent real numbers. This well-written, 94-page book by Khinchin covers the basic facts about this correspondence as well as some applications in diophantine approximation and measure-theoretic questions about c.f.. The book was published first in 1935, the later editions (1949, 1961) without significant changes. Though probably many things have been discovered since then, the text provides a good entry to this field of study.
Content
There are three chapters. The first one gives some basic and very useful (in-)equalities.
The second chapter starts out with proving, that every real number x has a (under a reasonable convention) unique c.f. converging to it; the rationals being exactly those whose
c.f. is finite. Later in this chapter we see also a proof of Lagrange's theorem: The quadratic irrationals are exactly those with a periodic c.f.
The main topic though are inequalities of form |x-p/q| < f(q), where x is a real, p,q integers and f a positive function; here one asks for solutions in p, q. This formula occurs for example if one wants to estimate how well x is approximated by an initial segment of the c.f. representing x. It is proved that the best one can get is |x-p_k/q_k| < 1/q_k^2; p_k/q_k being the value of the initial segment of length k of the c.f. representing x.
Rearranging this inequality one enters the field of diophantine approximation. Since c.f. are in some well-defined sense optimal in approximating reals, they are a good tool to study such problems. We see a theorem of Chebyshev's on the number of solutions in p,q\in Z for |xq-p-a|<3/q (a arbitrary). A non-pigeonhole-principle proof of Dirichlet's theorem is given, based on an inequality of chapter one.
The third chapter is concerned with the size of sets of reals defined by some property of their c.f. expansion. For example, a criterion on f is given such that the set of reals x with infinitely many solutions of |x-p/q| < f(q)/q in p,q\in Z is the complement of a null set. Another big result is Kuzmin's theorem. While this itself is a rather technical statement, it can be used to answer questions like: What roughly is the measure of the set of reals in the interval [0,1] whose c.f. has a 5 at the 2018th place? And with some further work: how often on average does a given number n occur in the c.f. of a given real? In fact, it is proved, that outside a set of measure zero, this density does only depend on n, but not on the real! There is no proof of the existence/value of Lévy's constant (nor Khinchin's constant), but its existence is mentioned and references are given.
Style
The text (the translation) is a pleasure to read. The general direction of enquiry and the significance of particular theorems is discussed. Every once in a while there is a remark about the history of the problem at hand. There are no exercises. There is a one-page index, but no bibliography. But the text is short, so neither is a real issue. The book is elementary: the only "tools" needed are sequences, series and some common integrals. For chapter three you will need the definition and basic properties of the Lebesgue-measure. The proofs in that chapter are also considerably longer and more technical.
February 13, 2009
Continued fractions are AWESOME! And Khinchin's text is well written and easy (and fun) to read.
February 4, 2008
Very clearly written, a definite classic. For a more up-to-date treatment, try the book of the same name by Rockett and Szusz.
July 29, 2015
Very short but interesting. Does proofs and other things pertaining to Continued Fractions.
Displaying 1 - 4 of 4 reviews