Lectures on Diophantine Approximations Part 1: g-adic numbers and Roth's theorem

LECTURES ON DIOPHANTINE APPROXIMATIONS Part 1: g-adic numbers and Roth's theorem......Page 1
Copyright......Page 2
Introduction......Page 5
LIST OF NOTATIONS......Page 7
LIST OF CONTENTS......Page 9
Part 1: p-adic and g-adic Numbers, and Their Approximations......Page 12
I. Valuations and pseudo-valuations......Page 14
1. Valuations and pseudo-valuations......Page 15
2. The p-adic valuations of Γ......Page 16
3. A further example......Page 17
4. Valuations and pseudo-valuations derived from given ones......Page 18
5. Bounded sequences, fundamental sequences, and null sequences......Page 20
6. The ring {K}_w and the ideal p......Page 22
7. The residue class ring K_w......Page 23
9. The limit notation......Page 24
10. The continuation of w(a) onto K_w......Page 25
11. The elements of K lie dense in K_w......Page 26
12. Fundamental sequences in K_w......Page 27
13. Equivalence of valuations and pseudo-valuations......Page 28
14. The valuations and pseudo-valuations of Γ......Page 29
15. Independent pseudo-valuations......Page 31
16. The decomposition theorem......Page 32
17. Convergent infinite series......Page 35
II. The p-adic, g-adic, and g*-adic series......Page 37
1. Notation......Page 38
2. The ring I_g and the ideal g......Page 40
3. The residue class ring I_g/g......Page 41
4. Systems of representatives......Page 42
5. Series for g-adic numbers......Page 43
6. Series for g*-adic numbers......Page 47
7. Sequences that converge with respect to all valuations of Γ......Page 51
III. A test for algebraic or transcendental numbers......Page 52
2. The minimum polynomial of an algebraic number......Page 53
3. An algebraic identity......Page 54
4. Inequalities for algebraic numbers......Page 56
5. A theorem on linear forms......Page 59
6. On a system of both real and p-adic linear forms......Page 61
7. Polynomials F(x) for which ω {F (a)} is small......Page 64
8. A necessary and sufficient condition for transcendency......Page 66
1. The continued fraction algorithm in the real case......Page 69
2. The convergents of the continued fraction for a_0......Page 70
3. The distinction between rational and irrational numbers......Page 71
4. Inequalities for |Q_ka_0 - P_k|......Page 73
6. The rational approximations of g-adic integers......Page 74
7. The continued fraction algorithm for a g-adic integer......Page 75
8. Two numerical examples......Page 78
10. The continued fraction algorithm for g*-adic numbers......Page 80
Part 2: Rational Approximations of Algebraic Numbers. The Problem and Its History......Page 84
2. Linear dependence and independence......Page 88
3. Generalized Wronski determinants......Page 89
4. The case of functions of one variable......Page 90
5. The general case......Page 91
6. An identity......Page 93
7. Majorants for U, V, and W......Page 96
8. The index of a polynomial......Page 98
9. The upper bound Θ_m(a; H_1 , . . . , H_m; r_1 , . . . , r_m)......Page 100
10. An upper bound for Θ_1(a; r; H)......Page 101
12. A recursive inequality for Θ_m. I......Page 103
13. A recursive inequality for Θ_m. II......Page 104
14. Proof of Roth's Lemma......Page 107
2. The powers of an algebraic number......Page 109
3. A lemma by Schneider......Page 110
4. The construction of A(x_1 , . . . , x_m). I......Page 112
5. The construction of A(x_1 , . . . , x_m). II......Page 113
6. The construction of A(x_1 , . . . , x_m). III......Page 115
1. The properties A_d, B, and C......Page 118
2. The selection of the parameters......Page 120
3. Application of Theorems 1 and 2......Page 123
4. Upper bounds for |A_{(1)}|......Page 124
5. An upper bound for |A_{(1)}|_g......Page 127
6. An upper bound for |D_{(1)}|......Page 128
7. Lower bounds for |N_{(1)}|......Page 131
8. Conclusion of the proof of the Main Lemma......Page 133
9. The first form of the First Approximation Theorem......Page 134
10. Polynomials in a field with a valuation......Page 136
11. Two applications of Lemma 1......Page 138
12. The property A'_d......Page 141
13. The second form of the First Approximation Theorem......Page 142
1. The two forms of the theorem......Page 144
2. The Theorem (2,II) implies the Theorem (2,I)......Page 145
3. The Theorem (2,I) implies the Theorem (2,II)......Page 146
4. The Theorem (2,I) implies the Theorem (1,I)......Page 148
5. The integers e_j......Page 151
6. The numbers g, g', g", p, σ, λ, μ......Page 154
7. The Theorem (1,Ι) implies the Theorem (2,Ι)......Page 156
1. The theorems of Roth and Ridout......Page 158
3. The powers of a rational number......Page 161
4. The equation P^(k) + Q^(k) + R^(k) = 0......Page 166
5. The approximation by rational integers......Page 169
6. An example......Page 172
Appendix A. Another proof of a lemma by Schneider......Page 174
Appendix B. A theorem by M. Cugiani......Page 180
Appendix C. The Approximation Theorems over Algebraic Number......Page 192