This monograph is intended to be a complete treatment of the metrical theory 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.
1100014490
Metrical Theory of Continued Fractions
This monograph is intended to be a complete treatment of the metrical theory 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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Metrical Theory of Continued Fractions
383
Metrical Theory of Continued Fractions
383Paperback(Softcover reprint of hardcover 1st ed. 2002)
$109.99
109.99
In Stock
Product Details
| ISBN-13: | 9789048161300 |
|---|---|
| Publisher: | Springer Netherlands |
| Publication date: | 12/08/2010 |
| Series: | Mathematics and Its Applications , #547 |
| Edition description: | Softcover reprint of hardcover 1st ed. 2002 |
| Pages: | 383 |
| Product dimensions: | 6.30(w) x 9.45(h) x 0.03(d) |
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