Diophantine Approximation (eBook)

CONTENTS 6
PREFACE 8
THE MATHEMATICAL WORK OF WOLFGANG SCHMIDT 9
Introduction 9
1 Geometry of numbers 9
2 Uniform distribution 10
3 Approximation of real numbers 11
4 Heights 12
5 Approximation of algebraic numbers by rationals 12
6 Norm form equations 14
7 Transcendental numbers 15
8 Riemann hypothesis for curves 16
9 Nonlinear approximation of real numbers 17
10 Zeros and small values of forms 18
11 Quadratic geometry of numbers 19
12 Approximation of algebraic numbers – quantitative results 19
13 Norm form equations – quantitative results 20
14 Linear recurrence sequences 21
Publications byW. Schmidt 22
Additional cited references 27
SCHÄFFER’S DETERMINANT ARGUMENT 29
1 Introduction 29
2 Proofs of Theorems 2 and 3 31
3 A lemma with four alternatives 38
4 Proof of Theorem 1 43
References 47
ARITHMETIC PROGRESSIONS AND TIC- TAC- TOE GAMES 48
1 Van der Waerden’s theorem 48
2 Hypercube Tic-Tac-Toe and positional games 53
3 Win vs. Weak Win 63
4 Old lower bounds 66
5 New lower bound results 72
6 More new lower bounds via games 76
7 Big Game–Small Game decomposition 85
8 How good are the new lower bounds? Strong Draw and Weak Win 91
References 99
METRIC DISCREPANCY RESULTS FOR SEQUENCES {nk x} AND DIOPHANTINE EQUATIONS 101
1 Introduction 101
2 Comments on conditions B, C and G 107
References 110
MAHLER’S CLASSIFICATION OF NUMBERS COMPARED WITH KOKSMA’S, II 112
1 Introduction 112
2 Results 113
3 An auxiliary result 116
4 The inductive construction 117
5 Completion of the proof of Theorem 2 122
6 Proof of Theorem 3 123
7 Proof of Theorem 4 124
References 126
RATIONAL APPROXIMATIONS TO A q-ANALOGUE OF p AND SOME OTHER q-SERIES 127
1 Introduction 127
2 Main results and reduction 128
3 Hypergeometric construction 130
4 Integral construction 135
5 Proofs 137
References 142
ORTHOGONALITY AND DIGIT SHIFTS IN THE CLASSICAL MEAN SQUARES PROBLEM IN IRREGULARITIES OF POINT DISTRIBUTION 144
1 Introduction 144
2 Linear distributions 146
3 Deduction of Theorem 1 149
4 Deduction of Theorem 2 150
5 Walsh functions 151
6 More weights and metrics 153
7 Approximation of the discrepancy function 153
8 Deduction of Theorem 5 158
9 Deduction of Theorems 3 and 4 159
References 161
APPLICATIONS OF THE SUBSPACE THEOREM TO CERTAIN DIOPHANTINE PROBLEMS 163
Introduction 163
The quotient problem 164
The d-th root problem 169
Integral points on certain affine varieties 171
References 175
A GENERALIZATION OF THE SUBSPACE THEOREM WITH POLYNOMIALS OF HIGHER DEGREE 177
1 Introduction 177
2 Twisted heights 181
3 Proof of Theorem 2.1 183
4 Height estimates 188
5 Proof of Theorem 1.3 193
References 199
ON THE DIOPHANTINE EQUATION Gn(x) = Gm(y) WITH Q(x, y) = 0 201
1 Introduction 201
2 Results 203
3 Proof of Theorem 1 206
4 Proof of Theorem 2 210
References 211
A CRITERION FOR POLYNOMIALS TO DIVIDE INFINITELY MANY k-NOMIALS 212
1 Introduction 212
2 The main results 213
3 Basic lemmas 215
4 Proofs 216
References 221
APPROXIMANTS DE PADÉ DES q-POLYLOGARITHMES 222
1 Introduction 222
2 Démonstration du Théorème 2 225
3 Confluence du Théorème 2 vers le Théorème 1 228
Références 231
THE SET OF SOLUTIONS OF SOME EQUATION FOR LINEAR RECURRENCE SEQUENCES 232
References 236
COUNTING ALGEBRAIC NUMBERS WITH LARGE HEIGHT I 237
References 243
CLASS NUMBER CONDITIONS FOR THE DIAGONAL CASE OF THE EQUATION OF NAGELL AND LJUNGGREN 244
1 Introduction 244
2 Cyclotomic fields 246
3 Classical results revisited proofs of Theorems 1 and 2
4 General upper bounds 258
5 Lower bounds and proof of Theorem 4 269
6 Conclusion 271
References 272
CONSTRUCTION OF APPROXIMATIONS TO ZETA- VALUES 273
1 Introduction 273
2 Common denominator for coefficients of Ak(z) 276
3 Upper bounds for the coefficients of Ak(x) 282
4 Some examples 287
References 291
QUELQUES ASPECTS DIOPHANTIENS DES VARIÉTÉS TORIQUES PROJECTIVES 292
1 Introduction et résultats 292
2 Généralités sur les variétés toriques projectives 296
3 Équations et indices d’obstruction successifs 301
4 Volumes, hauteurs d’espaces tangents et degrés 307
5 Un théorème de Bézout pour les poids de Chow 310
6 Hauteur normalisée 315
7 Optimalité du théorème des minimums algébriques successifs 320
8 Poids de Chow et hauteur des diviseurs monomiaux 325
Références 334
UNE INÉGALITÉ DE LOJASIEWICZ ARITHMÉTIQUE 336
1 Résultat 336
2 Hauteurs 337
3 Minimum local 339
4 Minimum sur un pavé 341
5 Conclusion 342
Références 342
ON THE CONTINUED FRACTION EXPANSION OF A CLASS OF NUMBERS 343
1 Introduction 343
2 Notation and statements of the main results 344
3 Proof of Theorem 2.1 346
4 Proof of Theorem 2.2 348
5 Proof of Theorem 2.3 352
6 Proof of Theorem 2.4 355
References 357
THE NUMBER OF SOLUTIONS OF A LINEAR HOMOGENEOUS CONGRUENCE 358
References 365
A NOTE ON LYAPUNOV THEORY FOR BRUN ALGORITHM 366
1 Introduction 366
2 A skew product 368
3 Brun algorithm 370
References 374
ORBIT SUMS AND MODULAR VECTOR INVARIANTS 375
1 Introduction 375
2 Orbit sums 380
3 Proof of Theorem 1 and Corollary 2 399
4 A universal invariant 401
5 Proof of Theorem 3 402
References 406
NEW IRRATIONALITY RESULTS FOR DILOGARITHMS OF RATIONAL NUMBERS 407
1 Introduction 407
2 Double integrals and permutation groups related to the dilogarithm 408
3 Irrationality results for Li2(r/s) 412
4 Concluding remarks 416
References 416