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Diophantine Approximation Festschrift for Wolfgang Schmidt

:	Hans Peter Schlickewei, Klaus Schmidt, Robert F. Tichy
:	Robert F. Tichy, Hans Peter Schlickewei, Klaus D. Schmidt
:	Diophantine Approximation Festschrift for Wolfgang Schmidt
:	Springer-Verlag
:	9783211742808
:	1
:	CHF 85.30
:

:	Arithmetik, Algebra
:	English

:	422
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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