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Continued Fractions - Andrew M. Rockett

Hardcover

Published: 8th August 1992
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This book presents the arithmetic and metrical theory of regular continued fractions and is intended to be a modern version of A. Ya. Khintchine's classic of the same title. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and applications are included (the continued fraction of "e", Ostrowski representations and "t"-expansions, period lengths of quadratic surds, the general Pell's equation, homogeneous and inhomogeneous diophantine approximation, Hall's theorem, the Lagrange and Markov spectra, asymmetric approximation, etc). Suitable for upper level undergraduate and beginning graduate students, the presentation is self-contained and the metrical results are developed as strong laws of large numbers.

"There are just a few, mostly aged books on continued fractions available. This one promises to have an enriching and stimulating effect on the way the subject presents itself to students and professional mathematicians alike." Ch. Baxa, Wien Monatshefte fur Mathematik, 1993

Preface
Notations
Introductionp. 1
What is a continued fraction?p. 1
Regular continued fractionsp. 3
The transformation T(x) = {1/x}p. 8
The quantity [actual symbol not reproducible]p. 9
Convergents to a number and its reciprocalp. 10
The ratio [actual symbol not reproducible]p. 10
The golden ratio and the Fibonacci sequencep. 11
The continued fraction for ep. 13
The Law of Best Approximationp. 19
Best approximationp. 19
The first proofp. 20
A theorem of Lagrangep. 22
Ostrowski's algorithm and a second proofp. 23
The approximation [actual symbol not reproducible]p. 27
The t-expansion of a real numberp. 32
Periodic Continued Fractionsp. 39
The classical theoremsp. 39
Period lengthsp. 49
Second order linear recurrencesp. 53
Applicationsp. 59
Gear ratio problemsp. 59
Pell's equationp. 64
Fermat's theorem on the sum of two squaresp. 69
Hall's theoremp. 72
A theorem of Hurwitzp. 79
The Lagrange and Markov spectrap. 83
Asymmetric approximation and Segre's theoremp. 110
Approximation by non-convergentsp. 112
Inhomogeneous approximationp. 116
Szekeres' empty parallelogram theoremp. 130
Metrical Theoryp. 137
Numbers with bounded partial quotientsp. 138
The Borel-Cantelli lemmap. 140
Random variables and expectationsp. 141
Chebyshev's inequality and large number lawsp. 143
The Gauss-Kuzmin theoremp. 151
The distribution of [actual symbol not reproducible]p. 155
Partial quotients are weakly dependentp. 159
Khintchine's theoremp. 160
The Khintchine-Levy theorem for [actual symbol not reproducible]p. 163
Applications to Metrical Diophantine Approximationp. 169
Bibliographyp. 177
Symbolsp. 181
Indexp. 185
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9789810210472
ISBN-10: 9810210477
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 196
Published: 8th August 1992
Country of Publication: SG
Dimensions (cm): 22.91 x 15.19  x 1.27
Weight (kg): 0.44