Metrical Theory of Continued Fractions

by Iosifescu, Marius; Kraaikamp, Cor

• ISBN13: 9781402008924
• ISBN10: 1402008929
• Format: Hardcover
• Copyright: 2002-10-01
• Publisher: Kluwer Academic Pub

Summary

The book is essentially based on recent work of the authors. In order to unify and generalize the results obtained so far, new concepts have been introduced, e.g., an infinite order chain representation of the continued fraction expansion of irrationals, the conditional measures associated with, and the extended random variables corresponding to that representation. Also, such procedures as singularization and insertion allow to obtain most of the continued fraction expansions related to the regular continued fraction expansion.The authors present and prove with full details for the first time in book form, the most recent developments in solving the celebrated 1812 Gauss' problem which originated the metrical theory of continued fractions. At the same time, they study exhaustively the Perron-Frobenius operator, which is of basic importance in this theory, on various Banach spaces including that of functions of bounded variation on the unit interval.The book is of interest to research workers and advanced Ph.D. students in probability theory, stochastic processes and number theory.

Table of Contents

Preface	ix
Frequently Used Notation	xv
Basic properties of the continued fraction expansion	1	(52)
A generalization of Euclid's algorithm	1	(13)
The continued fraction transformation τ	1	(3)
Continuants and convergents	4	(7)
Some special continued fraction expansions	11	(3)
Basic metric properties	14	(11)
Defining random variables of interest	14	(1)
Gauss' problem and measure	15	(2)
Fundamental intervals, and applications	17	(8)
The natural extension of τ	25	(28)
Definition and basic properties	25	(2)
Approximation coefficients	27	(4)
Extended random variables	31	(5)
The conditional probability measures	36	(3)
Paul Levy's solution to Gauss' problem	39	(4)
Mixing properties	43	(10)
Solving Gauss' problem	53	(112)
Banach space preliminaries	53	(3)
A few classical Banach spaces	53	(2)
Bounded essential variation	55	(1)
The Perron--Frobenius operator	56	(23)
Definition and basic properties	56	(6)
Asymptotic behaviour	62	(2)
Restricting the domain of the Perron--Frobenius operator	64	(6)
A solution to Gauss' problem for probability measures with densities	70	(1)
Computing variances of certain sums	71	(8)
Wirsing's solution to Gauss' problem	79	(22)
Elementary considerations	79	(6)
A functional-theoretic approach	85	(10)
The case of Lipschitz densities	95	(6)
Babenko's solution to Gauss' problem	101	(19)
Preliminaries	101	(2)
A symmetric linear operator	103	(8)
An `exact' Gauss--Kuzmin--Levy theorem	111	(8)
ψ-mixing revisited	119	(1)
Extending Babenko's and Wirsing's work	120	(15)
The Mayer--Roepstorff Hilbert space approach	120	(7)
The Mayer--Roepstorff Banach space approach	127	(3)
Mayer--Ruelle operators	130	(5)
The Markov chain associated with the continued fraction expansion	135	(30)
The Perron--Frobenius operator on BV (I)	135	(4)
An upper bound	139	(12)
Two asymptotic distributions	151	(5)
A generalization of a result of A. Denjoy	156	(9)
Limit theorems	165	(54)
Preliminaries	165	(4)
The Poisson law	169	(10)
The case of incomplete quotients	169	(2)
The case of associated random variable	171	(2)
Some extreme value theory	173	(6)
Normal convergence	179	(17)
Two general invariance principles	179	(3)
The case of incomplete quotients	182	(6)
The case of associated random variables	188	(8)
Convergence to non-normal stable laws	196	(17)
The case of incomplete quotients	196	(6)
Sums of incomplete quotients	202	(5)
The case of associated random variables	207	(6)
Fluctuation results	213	(6)
The case of incomplete quotients	213	(2)
The case of associated random variables	215	(4)
Ergodic theory of continued fractions	219	(94)
Ergodic theory preliminaries	219	(6)
A few general concepts	219	(5)
The special case of the transformations τ and τ	224	(1)
Classical results and generalizations	225	(32)
The case of incomplete quotients	225	(15)
Empirical evidence, and normal continued fraction numbers	240	(4)
The case of associated and extended random variables	244	(13)
Other continued fraction expansions	257	(24)
Preliminaries	257	(3)
Semi-regular continued fraction expansions	260	(4)
The singularization process	264	(2)
S-expansions	266	(7)
Ergodic properties of S-expansions	273	(8)
Examples of S-expansions	281	(18)
Nakada's α-expansions	281	(8)
Minkowski's diagonal continued fraction expansion	289	(3)
Bosma's optimal continued fraction expansion	292	(7)
Continued fraction expansions with σ-finite, infinite invariant measure	299	(14)
The insertion process	299	(1)
The Lehner and Farey continued fraction expansion	300	(7)
The backward continued fraction expansion	307	(6)
Appendix 1: Spaces, functions, and measures	313	(8)
A1.1	313	(1)
A1.2	313	(1)
A1.3	314	(1)
A1.4	314	(2)
A1.5	316	(3)
A1.6	319	(2)
Appendix 2: Regularly varying functions	321	(4)
A2.1	321	(2)
A2.2	323	(1)
A2.3	324	(1)
Appendix 3: Limit theorems for mixing random variables	325	(8)
A3.1	325	(2)
A3.2	327	(1)
A3.3	328	(5)
Notes and Comments	333	(14)
References	347	(30)
Index	377