Applications of Diophantine Approximation to Integral Points and Transcendence (Cambridge Tracts in Mathematics Book 212)

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. 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 1. Diophantine Approximation And Diophantine Equations -- 1.1. The Origins -- 1.2. From Thue To Roth -- 1.3. Exercises -- 1.4. Notes -- 2. Schmidt's Subspace Theorem And S-unit Equations -- 2.1. From Roth To Schmidt -- 2.2. The S-unit Equation -- 2.3. S-unit Points On Algebraic Varieties -- 2.4. Norm-form Equations -- 2.5. Exercises -- 2.6. Notes -- 3. Integral Points On Curves And Other Varieties -- 3.1. General Notions On Integral Points -- 3.2. The Chevalley -- Weil Theorem -- 3.3. Integral Points On Curves: Siegel's Theorem -- 3.4. Another Approach To Siegel's Theorem -- 3.5. Varieties Of Higher Dimension -- 3.6. Quadratic-integral Points On Curves -- 3.7. Rational Points -- 3.8. The Hilbert Irreducibility Theorem -- 3.9. Constructing Integral Points On Certain Surfaces -- 3.10. Exercises -- 3.11. Notes -- 4. Diophantine Equations With Linear Recurrences -- 4.1. Linear Recurrences -- 4.2. Zeros Of Recurrences -- 4.3. Quotients Of Recurrences And Gcd Estimates -- 4.4. Applications Of Gcd Estimates -- 4.5. Further Diophantine Problems With Recurrences -- 4.6. Fractional Parts Of Powers -- 4.7. Markov Numbers -- 4.8. Exercises -- 4.9. Notes -- 5. Some Applications Of The Subspace Theorem In Transcendental Number Theory -- 5.1. Transcendence Of Lacunary Series -- 5.2. Complexity Of Algebraic Numbers. Pietro Corvaja, Umberto Zannier. Includes Bibliographical References And Index. This introduction to Diophantine approximation and Diophantine equations, with applications to related topics, pays special regard to Schmidt's subspace theorem. It contains a number of results, some never before published in book form, and some new. The authors introduce various techniques and open questions to guide future research.