## Metrical Theory of Continued FractionsThis monograph is intended to be a complete treatment of the metrical the ory of the (regular) continued fraction expansion and related representations of real numbers. We have attempted to give the best possible results known so far, with proofs which are the simplest and most direct. The book has had a long gestation period because we first decided to write it in March 1994. This gave us the possibility of essentially improving the initial versions of many parts of it. Even if the two authors are different in style and approach, every effort has been made to hide the differences. Let 0 denote the set of irrationals in I = [0,1]. Define the (reg ular) continued fraction transformation T by T (w) = fractional part of n 1/w, w E O. Write T for the nth iterate of T, n E N = {O, 1, ... }, n 1 with TO = identity map. The positive integers an(w) = al(T - (W)), n E N+ = {1,2··· }, where al(w) = integer part of 1/w, w E 0, are called the (regular continued fraction) digits of w. Writing . for arbitrary indeterminates Xi, 1 :::; i :::; n, we have w = lim [al(w),··· , an(w)], w E 0, n--->oo thus explaining the name of T. The above equation will be also written as w = lim [al(w), a2(w),···], w E O. |

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### Contents

III | 1 |

IV | 4 |

V | 11 |

VI | 14 |

VII | 15 |

VIII | 17 |

IX | 25 |

X | 27 |

XLII | 182 |

XLIII | 188 |

XLIV | 196 |

XLV | 202 |

XLVI | 207 |

XLVII | 213 |

XLVIII | 215 |

XLIX | 219 |

XI | 31 |

XII | 36 |

XIII | 39 |

XIV | 43 |

XV | 53 |

XVI | 55 |

XVII | 56 |

XVIII | 62 |

XIX | 64 |

XX | 70 |

XXI | 71 |

XXII | 79 |

XXIII | 85 |

XXIV | 95 |

XXV | 101 |

XXVI | 103 |

XXVII | 111 |

XXVIII | 119 |

XXIX | 120 |

XXX | 127 |

XXXI | 130 |

XXXII | 135 |

XXXIII | 139 |

XXXIV | 151 |

XXXV | 156 |

XXXVI | 165 |

XXXVIII | 169 |

XXXIX | 171 |

XL | 173 |

XLI | 179 |

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### Common terms and phrases

a-expansions absolutely convergent algorithm an)neN+ an+i Assume asymptotic relative Banach space bounded linear operator BV(I Clearly consider continued fraction expansion convergence rate Corollary cr-algebra defined denned denote derived digits Doeblin dynamical system easy to check eigenvalue equation equivalent ergodic dynamical system Euclid's algorithm exists finite following result fundamental interval G N+ Gauss Hence holds i e N+ incomplete quotients inequality integers l,oo Lebesgue measure Lemma linear operator losifescu n e N+ natural extension neN+ non-decreasing Note obtain Perron-Frobenius operator pr(B probability measure probability space proof is complete proved random variables RCF convergents RCF expansion real-valued regularly varying Remark Samur sequence singularization area slowly varying SRCF stochastic process strictly stationary Subsection trace class varying of index yields

### Popular passages

Page 358 - M. (1990) A survey of the metric theory of continued fractions, fifty years after Doeblin's 1940 paper. In:

Page 370 - A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21, 549—563.

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Page 354 - On the central limit theorem for random variables related to the continued fraction expansion. Colloq. Math. 71,

Page 362 - A Tribute to Emil Grosswald: Number Theory and Related Analysis. Contemporary Mathematics 143.

Page 349 - Metrical properties of some random variables connected with the continued fraction expansion. Indag. Math.

Page 348 - Baladi, V. and Keller, G. (1990) Zeta functions and transfer operators for piecewise