This introduction to the theory of Diophantine approximation pays special regard to Schmidt's subspace theorem and to its applications to Diophantine equations and related topics. The geometric viewpoint on Diophantine equations has been adopted throughout the book. It includes a number of results, some published here for the first time in book form, and some new, as well as classical material presented in an accessible way. Graduate students and experts alike will find the book's broad approach useful for their work, and will discover new techniques and open questions to guide their research. It contains concrete examples and many exercises (ranging from the relatively simple to the much more complex), making it ideal for self-study and enabling readers to quickly grasp the essential concepts.
Author(s): Pietro Corvaja, Umberto Zannier
Series: Cambridge Tracts in Mathematics 212
Publisher: Cambridge University Press
Year: 2018
Language: English
Pages: 209
Contents......Page 6
Preface......Page 7
Notation and Conventions......Page 10
Introduction......Page 12
1.1 The Origins......Page 14
1.2 From Thue to Roth......Page 25
1.3 Exercises......Page 36
1.4 Notes......Page 38
2.1 From Roth to Schmidt......Page 40
2.2 The S-Unit Equation......Page 43
2.3 S-Unit Points on Algebraic Varieties......Page 46
2.4 Norm-Form Equations......Page 49
2.5 Exercises......Page 53
2.6 Notes......Page 55
3.1 General Notions on Integral Points......Page 59
3.2 The Chevalley–Weil Theorem......Page 64
3.3 Integral Points on Curves: Siegel’s Theorem......Page 71
3.4 Another Approach to Siegel’s Theorem......Page 76
3.5 Varieties of Higher Dimension......Page 81
3.6 Quadratic-Integral Points on Curves......Page 100
3.7 Rational Points......Page 103
3.8 The Hilbert Irreducibility Theorem......Page 106
3.9 Constructing Integral Points on Certain Surfaces......Page 120
3.10 Exercises......Page 124
3.11 Notes......Page 127
4.1 Linear Recurrences......Page 130
4.2 Zeros of Recurrences......Page 134
4.3 Quotients of Recurrences and gcd Estimates......Page 137
4.4 Applications of gcd Estimates......Page 145
4.5 Further Diophantine Problems with Recurrences......Page 153
4.6 Fractional Parts of Powers......Page 164
4.7 Markov Numbers......Page 168
4.8 Exercises......Page 173
4.9 Notes......Page 178
5.1 Transcendence of Lacunary Series......Page 183
5.2 Complexity of Algebraic Numbers......Page 187
References......Page 199
Index......Page 208
