# Metrical Theory of Continued Fractions

Springer Science & Business Media, Sep 30, 2002 - Mathematics - 383 pages
This 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
 L 224 LI 225 LII 240 LIII 244 LIV 257 LV 260 LVI 264 LVII 266 LVIII 273 LIX 281 LX 289 LXI 292 LXII 299 LXIII 300 LXIV 307 LXV 313 LXVI 314 LXVIII 316 LXIX 319 LXX 321 LXXI 323 LXXII 324 LXXIII 325 LXXIV 327 LXXV 328 LXXVI 333 LXXVII 347 LXXVIII 377 Copyright

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