The subject of continued fractions is a staple in any good mathematics education. There are several decent introductory books on the topic. This book starts where those end.
The author’s goal is to summarize many of the recent advances in this topic and its applications. As a result, almost half of the references in the bibliography were published after 1990. Another consequence, though, is that the audience is limited. Without a good deal of familiarity with continued fractions, this text is somewhat inaccessible. This is somewhat unfortunate, since there are some great topics contained in the book.
The topics include algorithms for calculating greatest common denominators and continued fractions for arbitrary Euclidean domains (especially rings of integers in the complex field), Diophantine approximation, multidimensional Diophantine approximation, continued fraction Cantor sets, techniques using functional analysis and generating functions, and issues of convergence. The topics are well-researched and well presented — provided you have the necessary background. And while much of the math presented here is somewhat recent, even the older mathematics is given a new life with the help of modern computing abilities.
Donald L. Vestal is Assistant Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)sdstate.edu