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Applications of Diophantine approximation to integral points and transcendence. (English) Zbl 1452.11004

Cambridge Tracts in Mathematics 212. Cambridge: Cambridge University Press (ISBN 978-1-108-42494-3/hbk; 978-1-108-34809-6/ebook). x, 198 p. (2018).
This book presents (some) applications of Diophantine approximation to Diophantine equations. It is intended to be accessible to, e.g., graduate students, without specific prerequisites. The following subjects are covered.
Chapter 1 is devoted to classical statements (e.g., Pell’s equation, Thue’s theorem, Roth’s theorem), as well as results of Ridout-Mahler-Lang, and applications.
Chapter 2 addresses the subspace theorem (Schmidt/Schlickewei) and \(S\)-unit equations.
Chapter 3 studies integral points on curves and varieties (Siegel’s theorem, Hilbert’s irreducibility theorem, and Chevalley-Weil’s theorem).
Chapter 4 is devoted to linear recurrences, with several applications (from zeta functions of dynamical systems to properties of \(||\alpha^n|| = \min(\{\alpha^n\}, 1 - \{\alpha^n\})\), and to Markov numbers.
The last chapter gives applications of the subspace theorem to transcendence (lacunary series, (block-)complexity of algebraic numbers, automatic reals, automatic continued fractions).
The book also contains exercises and notes for each chapter. Finally a large bibliography is provided (while the authors explain in the introduction their choice to avoid trying to give an exhaustive list of references).
This book is definitely a nice and interesting contribution: it can be used to learn or to refresh what one should know on the subject, leading the reader – almost without effort – to master the bases and to arrive at quite recent results. A book that will certainly find a place in any good institutional or personal library.

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11Dxx Diophantine equations
11J13 Simultaneous homogeneous approximation, linear forms
11J20 Inhomogeneous linear forms
11J68 Approximation to algebraic numbers
11J81 Transcendence (general theory)
11J87 Schmidt Subspace Theorem and applications
11B85 Automata sequences
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