The Euclidean algorithm is one of the oldest in mathematics, while the study of continued fractions as tools of approximation goes back at least to Euler and Legendre. While our understanding of continued fractions and related methods for simultaneous diophantine approximation has burgeoned over the course of the past decade and more, many of the results have not been brought together in book form. Continued fractions have been studied from the perspective of number theory, complex analysis, ergodic theory, dynamic processes, analysis of algorithms, and even theoretical physics, which has further complicated the situation.

This book places special emphasis on continued fraction Cantor sets and the Hausdorff dimension, algorithms and analysis of algorithms, and multi-dimensional algorithms for simultaneous diophantine approximation. Extensive, attractive computer-generated graphics are presented, and the underlying algorithms are discussed and made available.

Contents:

• Generalizations of the gcd and the Euclidean Algorithm
• Continued Fractions with Small Partial Quotients
• Ergodic Theory
• Complex Continued Fractions
• Multidimensional Diophantine Approximation
• Powers of an Algebraic Integer
• Marshall Hall's Theorem
• Functional-Analytic Techniques
• The Generating Function Method
• Conformal Iterated Function Systems
• Convergence of Continued Fractions

View Full Text (10,116 KB)

Readership: Graduate students and researchers in pure and applied mathematics.

"The topics are well-researched and well presented ... even the older mathematics is given a new life with the help of modern computing abilities."