## Preface

### Abstract

The main justification for this book is that there have been significant advances in continued fractions over the past decade, but these remain for the most part scattered across the literature, and under the heading of topics from algebraic number theory to theoretical plasma physics. We now have a better understanding of the rate at which assorted continued fraction or greatest common denominator (gcd) algorithms complete their tasks. The number of steps required to complete a gcd calculation, for instance, has a Gaussian normal distribution. We know a lot more about badly approximable numbers. There are several related threads here. A badly approximable number is a number x such that {q|p − qx|: p, q ∈ Z and q ̸ = 0} is bounded below by a positive constant; badly approximable numbers have continued fraction expansions with bounded partial quotients, and so we are led to consider a kind of Cantor set EM consisting of all x ∈ [0, 1] such that the partial quotients of x are bounded above by M. The notion of a badly approximable rational number has the ring of crank mathematics, but it is quite natural to study the set of rationals r with partial quotients bounded by M. The number of such rationals with denominators up to n, say, turns out to be closely related to the Hausdorff dimension of EM, (comparable to n2dimEM) which is in turn related to the spectral radius of linear operators LM,s, acting on some suitably chosen space of functions f, and given by LM,sf(t) = ∑m k=1 (k + t) −sf(1/(k + t)). Similar operators have been studied by, among others, David Ruelle, in connection with theoretical one-dimensional plasmas, and they are related to entropy. Alongside these developments there has been a dramatic increase in the computational power available to investigators. This has been helpful on the theoretical side, as one is more likely to seek a proof for a result when,