Tichy / Schlickewei / Schmidt Diophantine Approximation

Festschrift for Wolfgang Schmidt
2008
ISBN: 978-3-211-74280-8
Verlag: Springer Wien
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Französisch, 422 Seiten, eBook

Reihe: Developments in Mathematics

This volume contains 21 research and survey papers on recent developments in the field of diophantine approximation, which are based on lectures given at a conference at the Erwin Schrödinger-Institute (Vienna, 2003). The articles are either in the spirit of more classical diophantine analysis or of a geometric or combinatorial flavor. Several articles deal with estimates for the number of solutions of diophantine equations as well as with congruences and polynomials.

Zielgruppe

Research

Autoren/Hrsg.

Schmidt, Klaus

Tichy, Robert F.

Schlickewei, Hans Peter

Inhaltsverzeichnis

1;CONTENTS;6
2;PREFACE;8
3;THE MATHEMATICAL WORK OF WOLFGANG SCHMIDT;9
3.1;Introduction;9
3.2;1 Geometry of numbers;9
3.3;2 Uniform distribution;10
3.4;3 Approximation of real numbers;11
3.5;4 Heights;12
3.6;5 Approximation of algebraic numbers by rationals;12
3.7;6 Norm form equations;14
3.8;7 Transcendental numbers;15
3.9;8 Riemann hypothesis for curves;16
3.10;9 Nonlinear approximation of real numbers;17
3.11;10 Zeros and small values of forms;18
3.12;11 Quadratic geometry of numbers;19
3.13;12 Approximation of algebraic numbers – quantitative results;19
3.14;13 Norm form equations – quantitative results;20
3.15;14 Linear recurrence sequences;21
3.16;Publications byW. Schmidt;22
3.17;Additional cited references;27
4;SCHÄFFER’S DETERMINANT ARGUMENT;29
4.1;1 Introduction;29
4.2;2 Proofs of Theorems 2 and 3;31
4.3;3 A lemma with four alternatives;38
4.4;4 Proof of Theorem 1;43
4.5;References;47
5;ARITHMETIC PROGRESSIONS AND TIC- TAC- TOE GAMES;48
5.1;1 Van der Waerden’s theorem;48
5.2;2 Hypercube Tic-Tac-Toe and positional games;53
5.3;3 Win vs. Weak Win;63
5.4;4 Old lower bounds;66
5.5;5 New lower bound results;72
5.6;6 More new lower bounds via games;76
5.7;7 Big Game–Small Game decomposition;85
5.8;8 How good are the new lower bounds? Strong Draw and Weak Win;91
5.9;References;99
6;METRIC DISCREPANCY RESULTS FOR SEQUENCES {nk x} AND DIOPHANTINE EQUATIONS;101
6.1;1 Introduction;101
6.2;2 Comments on conditions B, C and G;107
6.3;References;110
7;MAHLER’S CLASSIFICATION OF NUMBERS COMPARED WITH KOKSMA’S, II;112
7.1;1 Introduction;112
7.2;2 Results;113
7.3;3 An auxiliary result;116
7.4;4 The inductive construction;117
7.5;5 Completion of the proof of Theorem 2;122
7.6;6 Proof of Theorem 3;123
7.7;7 Proof of Theorem 4;124
7.8;References;126
8;RATIONAL APPROXIMATIONS TO A q-ANALOGUE OF p AND SOME OTHER q-SERIES;127
8.1;1 Introduction;127
8.2;2 Main results and reduction;128
8.3;3 Hypergeometric construction;130
8.4;4 Integral construction;135
8.5;5 Proofs;137
8.6;References;142
9;ORTHOGONALITY AND DIGIT SHIFTS IN THE CLASSICAL MEAN SQUARES PROBLEM IN IRREGULARITIES OF POINT DISTRIBUTION;144
9.1;1 Introduction;144
9.2;2 Linear distributions;146
9.3;3 Deduction of Theorem 1;149
9.4;4 Deduction of Theorem 2;150
9.5;5 Walsh functions;151
9.6;6 More weights and metrics;153
9.7;7 Approximation of the discrepancy function;153
9.8;8 Deduction of Theorem 5;158
9.9;9 Deduction of Theorems 3 and 4;159
9.10;References;161
10;APPLICATIONS OF THE SUBSPACE THEOREM TO CERTAIN DIOPHANTINE PROBLEMS;163
10.1;Introduction;163
10.2;The quotient problem;164
10.3;The d-th root problem;169
10.4;Integral points on certain affine varieties;171
10.5;References;175
11;A GENERALIZATION OF THE SUBSPACE THEOREM WITH POLYNOMIALS OF HIGHER DEGREE;177
11.1;1 Introduction;177
11.2;2 Twisted heights;181
11.3;3 Proof of Theorem 2.1;183
11.4;4 Height estimates;188
11.5;5 Proof of Theorem 1.3;193
11.6;References;199
12;ON THE DIOPHANTINE EQUATION Gn(x) = Gm(y) WITH Q(x, y) = 0;201
12.1;1 Introduction;201
12.2;2 Results;203
12.3;3 Proof of Theorem 1;206
12.4;4 Proof of Theorem 2;210
12.5;References;211
13;A CRITERION FOR POLYNOMIALS TO DIVIDE INFINITELY MANY k-NOMIALS;212
13.1;1 Introduction;212
13.2;2 The main results;213
13.3;3 Basic lemmas;215
13.4;4 Proofs;216
13.5;References;221
14;APPROXIMANTS DE PADÉ DES q-POLYLOGARITHMES;222
14.1;1 Introduction;222
14.2;2 Démonstration du Théorème 2;225
14.3;3 Confluence du Théorème 2 vers le Théorème 1;228
14.4;Références;231
15;THE SET OF SOLUTIONS OF SOME EQUATION FOR LINEAR RECURRENCE SEQUENCES;232
15.1;References;236
16;COUNTING ALGEBRAIC NUMBERS WITH LARGE HEIGHT I;237
16.1;References;243
17;CLASS NUMBER CONDITIONS FOR THE DIAGONAL CASE OF THE EQUATION OF NAGELL AND LJUNGGREN;244
17.1;1 Introduction;244
17.2;2 Cyclotomic fields;246
17.3;3 Classical results revisited; proofs of Theorems 1 and 2;255
17.4;4 General upper bounds;258
17.5;5 Lower bounds and proof of Theorem 4;269
17.6;6 Conclusion;271
17.7;References;272
18;CONSTRUCTION OF APPROXIMATIONS TO ZETA- VALUES;273
18.1;1 Introduction;273
18.2;2 Common denominator for coefficients of Ak(z);276
18.3;3 Upper bounds for the coefficients of Ak(x);282
18.4;4 Some examples;287
18.5;References;291
19;QUELQUES ASPECTS DIOPHANTIENS DES VARIÉTÉS TORIQUES PROJECTIVES;292
19.1;1 Introduction et résultats;292
19.2;2 Généralités sur les variétés toriques projectives;296
19.3;3 Équations et indices d’obstruction successifs;301
19.4;4 Volumes, hauteurs d’espaces tangents et degrés;307
19.5;5 Un théorème de Bézout pour les poids de Chow;310
19.6;6 Hauteur normalisée;315
19.7;7 Optimalité du théorème des minimums algébriques successifs;320
19.8;8 Poids de Chow et hauteur des diviseurs monomiaux;325
19.9;Références;334
20;UNE INÉGALITÉ DE LOJASIEWICZ ARITHMÉTIQUE;336
20.1;1 Résultat;336
20.2;2 Hauteurs;337
20.3;3 Minimum local;339
20.4;4 Minimum sur un pavé;341
20.5;5 Conclusion;342
20.6;Références;342
21;ON THE CONTINUED FRACTION EXPANSION OF A CLASS OF NUMBERS;343
21.1;1 Introduction;343
21.2;2 Notation and statements of the main results;344
21.3;3 Proof of Theorem 2.1;346
21.4;4 Proof of Theorem 2.2;348
21.5;5 Proof of Theorem 2.3;352
21.6;6 Proof of Theorem 2.4;355
21.7;References;357
22;THE NUMBER OF SOLUTIONS OF A LINEAR HOMOGENEOUS CONGRUENCE;358
22.1;References;365
23;A NOTE ON LYAPUNOV THEORY FOR BRUN ALGORITHM;366
23.1;1 Introduction;366
23.2;2 A skew product;368
23.3;3 Brun algorithm;370
23.4;References;374
24;ORBIT SUMS AND MODULAR VECTOR INVARIANTS;375
24.1;1 Introduction;375
24.2;2 Orbit sums;380
24.3;3 Proof of Theorem 1 and Corollary 2;399
24.4;4 A universal invariant;401
24.5;5 Proof of Theorem 3;402
24.6;References;406
25;NEW IRRATIONALITY RESULTS FOR DILOGARITHMS OF RATIONAL NUMBERS;407
25.1;1 Introduction;407
25.2;2 Double integrals and permutation groups related to the dilogarithm;408
25.3;3 Irrationality results for Li2(r/s);412
25.4;4 Concluding remarks;416
25.5;References;416

-Dedication to Wolfgang Tichy.-Schäffer´s Determinant Argument.-Arithmetic progressions and Tic-Tac-Toe games.-Metric discrepancy results for sequences {NkX } and Diophantine equations.-Mahler´s classification of numbers compared with Kosma´s, II.-Rational approximations to a q-analogue of p and some other q-series.-Orthogonality and digit shifts in the classical Mean Squares problem in irregularities of point distribution.-Applications of the Subspace Theorem to certain Diophantine problems.-A generalization of the Subspace Theorem with polynomials of higher degree.-On the Diophantine equation Gn (x) = Gm (y) with Q(x,y) = 0.-A criterion for polynomials to divide infinitely many k-nomials.-Approximants de Padê des q-Polylogarithmes.-The set of solutions of some equation for linear recurrence sequences.-Counting algebraic numbers with large height I.-Class number conditions for the diagonal case of the equation of Nagell-Ljunggren.-Construction of approximations to zeta-values.-Quelques aspects Diophantiens des variétés Toriques Projectives.-Une inégalité de Lojasiewicz arithmétique.-On the continued fraction expansion of a class of numbers.-The number of solutions of a linear homogeneous congruence.-A note on Lyapunov theory for Brun algorithm.-Orbit sums and modular vector invariants.-New irrationality results for dologarithms of rational numbers.