Metrical Theory of Continued Fractions

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Springer Science & Business Media, Sep 30, 2002 - Mathematics - 383 pages
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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
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IV
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V
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VIII
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LIII
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LIV
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LIX
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LXI
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LXXV
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LXXVI
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LXXVIII
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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.
Page 366 - G. (1992) The optimal zone for large deviations of the denominators of continued fractions.
Page 358 - A very simple proof of a generalization of the Gauss—Kuzmin—Lvy theorem on continued fractions, and related questions. Rev. Roumaine Math. Pures Appl. 37,
Page 361 - On the application of dependence with complete connections to the metrical theory of G-continued fractions. Lithuanian Math. J.
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

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