This very nice text offers an impressive and breathtaking overview of Diophantine approximation, Diophantine equations, and related classical topics. The book covers a wide range of material from very elementary results through the more serious standard topics.
The book opens by introducing material usually found in an undergraduate number theory book. The writing style here is, at times, lively and cute. The book then moves into the elementary aspects of Diophantine approximation (a la Dirichet, Kronecker, and Hurwitz), introduces Pade approximation, and develops the theory of continued fractions and the connection with the so-called "Pell's Equation". The text then treats some topics that are less common in such books: Apery's and Beukers' proofs of the irrationality of ζ(3) and the connection between factoring and continued fractions.
The book then takes on a much more serious tone as it moves through the geometry of numbers (including a discussion, but no proof of Minkowski's Successive Minima Theorem), elementary transcendence results, and Roth's Theorem (whose proof is included in detail). The text closes with a treatment of some more modern topics including the abc-conjecture and an introduction to p -adic analysis.
While the book has exercises throughout, the intended audience is not clear. The voice of the book varies dramatically — sometimes very basic notions are described in lively detail as if the reader might be beginning his or her journey into mathematics, but then on page 8 one of the standard proofs of the irrationality of π is offered devoid of any intuition whatsoever. Then after this technically complicated argument, on page 12 we are informed of the definition of the Division Algorithm. Certainly material on Roth's Theorem is not designed for the usual undergraduate mathematics student.
Despite this slight unevenness, for the professional mathematician this book provides a fine resource. I enjoyed it and am sure I will be a useful reference.
Edward B. Burger teaches at Williams College. He is the author of many books, including Exploring the Number Jungle: A Journey into Diophantine Analysis.