Continued Fractions 1st Edition by Andrew M Rockett (PDF)

Description

This book presents the arithmetic and metrical theory of regular continued fractions and is intended to be a modern version of A. Ya. Khintchine’s classic of the same title. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and applications are included (the continued fraction of e, Ostrowski representations and t-expansions, period lengths of quadratic surds, the general Pell’s equation, homogeneous and inhomogeneous diophantine approximation, Hall’s theorem, the Lagrange and Markov spectra, asymmetric approximation, etc). Suitable for upper level undergraduate and beginning graduate students, the presentation is self-contained and the metrical results are developed as strong laws of large numbers.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The part of the book that I’ve read most closely is the chapter on the measure theory of continued fractions. It by itself makes the book worth getting because it has material that seems only otherwise to be in Khinchin’s book, and it gives a more modern presentation of the measure theory involved than Khinchin gives.Here are some of the topics this book covers on the measure theory of continued fractions: The set of numbers with bounded partial quotients has measure 0, and moreover a set of numbers whose partial quotients grow slowly enough also has measure 0. On the other hand, a set of numbers whose partial quotients are very large infinitely often has measure 0. By a simple argument they bounded the probability that the nth partial quotient of a number has a particular value k, in terms of k, independently of n. They prove the Gauss-Kuzmin theorem, which gives an asymptotic for the probability that the nth remainder of a continued fraction is bigger than a certain value x, in terms of x and with an error term in terms of n. They prove that partial quotients are close to being independent. Finally they prove the Khinchin-Levy theorem. I haven’t read their proof of the Khinchin-Levy theorem, but the other proofs were not hard to digest.However, there are some infelicities that should be fixed if a new edition of the book is ever released. Why not mention that what the authors call on page 141 the “Beppo-Levi theorem” (giving the connotation that Beppo and Levi were two mathematicians rather than one mathematician’s first and last names) is commonly known as the monotone convergence theorem? If I hadn’t seen joint distributions before I wouldn’t be able to figure out in a preicse way what the authors mean on page 144; they define the joint distribution function using the as yet undefined expression P(f_1