Continued Fractions

The following sections are included:

• CONTENTS

• PREFACE

• NOTATIONS

The following sections are included:

• What is a continued fraction?

• Regular continued fractions

• The transformation T(x) = {1/x}

• The quantity D_k = B_kt − A_k

• Convergents to a number and its reciprocal

• The ratio ξ_k = B_{k − 1}/B_k

• The golden ratio and the Fibonacci sequence

• The continued fraction for e.

• NOTES

The following sections are included:

• Best approximations

• The first proof

• A theorem of Lagrange

• Ostrowski's algorithm and a second proof

• The approximation b‖bt‖ < K

• The t-expansion of a real number

• NOTES

The following sections are included:

• The classical theorems

• Period lengths

• Second order linear recurrences

• NOTES

The following sections are included:

• Gear ratio problems

• Pell's equation

• Fermat's theorem on the sum of two squares

• Hall's Theorem

• A theorem of Hurwitz

• The Lagrange and Markov spectra

• Asymmetric approximation and Segre's theorem

• Approximation by non-convergents

• Inhomogeneous approximation

• Szekeres' empty parallelogram theorem

• NOTES

The following sections are included:

• Numbers with bounded partial quotients

• The Borel-Cantelli lemma

• Random variables and expectations

• Chebyshev's inequality and large number laws

• The Gauss-Kuzmin theorem

• The distribution of ξ_n

• Partial quotients are weakly dependent

• Khintchine's theorem

• The Khintchine-Lévy theorem for

• NOTES

In Chapter II, we examined the solutions of the diophantine approximation b‖bt‖ < K for a constant K > 0 and, in Chapter IV, we saw that for numbers with bounded partial quotients, there were no solutions for sufficiently small values of K. As we now know from Chapter V, such numbers may be considered as exceptional cases and so we now shall study the solutions of b‖bt‖ < K for almost all t ∈ [0, 1). We begin with a result stated implicitly in F. Bernstein [1912]…

The following sections are included:

• BIBLIOGRAPHY

• SYMBOLS

• INDEX