Chudnovsky Additive Number Theory

Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson
1. Auflage 2010
ISBN: 978-0-387-68361-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 361 Seiten

This impressive volume is dedicated to Mel Nathanson, a leading authoritative expert for several decades in the area of combinatorial and additive number theory. For several decades, Mel Nathanson's seminal ideas and results in combinatorial and additive number theory have influenced graduate students and researchers alike. The invited survey articles in this volume reflect the work of distinguished mathematicians in number theory, and represent a wide range of important topics in current research.

Autoren/Hrsg.

Chudnovsky, David

Chudnovsky, Gregory

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;Addictive Number Theory;14
3.1;A True Story;14
3.2;Remarks on Some of My Articles;15
3.3;References;19
4;Sum-Product Theorems and Applications;22
4.1;Introduction;22
4.2;0 Sum-Product Theorem in Fp;23
4.3;1 Preliminaries from Additive Combinatorics;23
4.4;2 Some Tools from Graph Theory: The Balog--Szemerédi--Gowers Theorem;27
4.5;3 Exponential Sum Estimate;29
4.6;4 Additive Relations in Multiplicative Groups;34
4.7;5 Multilinear Exponential Sums;38
4.8;6 Extensions to 'Almost Groups'
;39
4.9;7 Sum-Product Theorem and Gauss Sums in Arbitrary Finite Fields;39
4.10;8 The Case of General Polynomial (mod p);40
4.11;9 The Sum-Product in Zq =Z/qZ;42
4.12;10 Exponential Sums in Finite Commutative Rings;44
4.13;11 Euclidean Algorithm in Algebraic Number Fields;45
4.14;12 Application to QUE;47
4.15;References;50
5;Can You Hear the Shape of a Beatty Sequence?;52
5.1;1 Introduction;52
5.2;2 Proofs;55
5.2.1;2.1 Proof of Theorem 1;55
5.2.2;2.2 Proof of Theorem 2;56
5.2.3;2.3 Proof of Theorem 3;58
5.2.4;2.4 Rasmussen's Approach to Conjecture 1;60
5.2.5;2.5 Proof of Theorem 4;62
5.3;3 Open Questions Concerning Generalized Polynomials;64
5.4;References;65
6;Variance of Signals and Their Finite Fourier Transforms;66
6.1;1 Eigenvalue and Eigenvectors of the Finite Fourier Matrix;66
6.1.1;1.1 McClellan Basis;68
6.1.2;1.2 Carlitz/Morton Basis;69
6.1.3;1.3 Dickinson--Steiglitz or Hofstatder Basis;70
6.2;2 Discrete Analogs;72
6.3;3 Theta Function Expressions for the Fourier Eigenvectors;72
6.4;4 Variational Principles for the Determination of Eigenfunctions of the Discrete Fourier Transform
;76
6.5;5 Discrete Uncertainty Principle;79
6.6;6 Explicit expressions for the matrix Mw2 in the Discrete
Version of the Uncertainty Principle
;80
6.7;7 Theta Function Bounds for Minimal Eigenvalues
in the Discrete Uncertainty Principle
;82
6.8;8 Numerical Evaluation of the Minimal Eigenvalues in the Discrete Uncertainty Principle
;85
6.9;9 Extensions of the Heisenberg-Weyl Inequality in the Continuous and Discrete Cases
;85
6.9.1;9.1 The Dickenson-Steiglitz Basis as Derived from Variational Principles
;88
6.10;References;88
7;Sparse Sets in Time and Frequency Related to Diophantine Problems and Integrable Systems;90
7.1;The Hilbert Matrix and Related Operators;90
7.2;General Prolate Functions;91
7.3;General Prolate Functions and Commuting Differential Operators
;93
7.4;Szego Problem and Concentrated Polynomials
;94
7.5;Hilbert Matrix and a Commuting Differential Operator;96
7.6;Szego Problem and Arbitrary Unions of Intervals
;97
7.7;Garnier Isomonodromy Deformation Equations;98
7.8;Explicit Expressions for the Non-Linear Darboux Transform;100
7.9;Darboux Transformation for m = 3 Case
;103
7.10;Generalized Prolate Functions and Another Isomonodromy Problem
;104
7.11;Generalized Prolate Matrices;104
7.12;Eigenvalue Problems for Hankel Matrices and Fourth Order Differential Equations
;105
7.13;Variational Principles and q: Difference Equations;108
7.14;References;110
8;Addition Theorems in Acyclic Semigroups;112
8.1;1 Introduction;112
8.2;2 Cayley Graphs on Semigroups;113
8.3;3 Vosper's Theorem;116
8.4;References;117
9;Small Sumsets in Free Products of z/2z
;118
9.1;1 Introduction;118
9.2;2 The Function kG(r,s)
;119
9.3;3 Free products of groups;120
9.4;4 Proof of m G(r,s) k G(r,s)
;121
9.5;5 Optimality;124
9.6;6 Proof of m G(r,s) k G(r,s)
;125
9.7;References;126
10;A Combinatorial Approach to Sums of Two Squares and Related Problems;127
10.1;1 Introduction;127
10.1.1;1.1 The Sums of Two Squares Theorem;127
10.1.2;1.2 Zagier's Proof;130
10.1.3;1.3 Heath-Brown's Proof
;130
10.1.4;1.4 Grace' Lattice Point Proof;131
10.1.5;1.5 Lucas' Work on Regular Satins;132
10.1.6;1.6 A Short Proof;132
10.1.6.1;1.6.1 The Long Version;132
10.1.6.2;1.6.2 A Short Version of the Proof;135
10.2;2 How Zagier's Involution can be Motivated;136
10.2.1;2.1 First Motivaton;136
10.2.2;2.2 Making the Proof Constructive;138
10.2.3;2.3 A Motivation Due to Dijkstra;139
10.2.4;2.4 Comparison;140
10.3;3 Generalization of the Method;141
10.3.1;3.1 d = 0
;143
10.3.2;3.2 d = 1
;143
10.3.3;3.3 d = 2
;143
10.3.3.1;3.3.1 The Case p=x2 +2yz;143
10.3.3.2;3.3.2 Proof of Theorem 3;145
10.3.4;3.4 d = 3
;147
10.3.4.1;3.4.1 The Case p = 3x2+4y2
;147
10.3.4.2;3.4.2 Proof of Theorem 4;148
10.3.5;3.5 d = 4
;149
10.4;4 On Infinite but Incomplete Mappings
;149
10.5;References;151
11;A Note on Elkin's Improvement of Behrend's Construction;153
11.1;1 Introduction;153
11.2;2 The Proof;154
11.3;3 A Question of Graham;156
11.4;References;156
12;Distinct Matroid Base Weights and Additive Theory;157
12.1;1 Introduction;157
12.2;2 Terminology and Preliminaries;160
12.3;3 Proof of the Main Result;161
12.4;References;162
13;The Postage Stamp Problem and Essential Subsets in Integer Bases;164
13.1;1 Essential Subsets of Integer Bases;164
13.2;2 The Postage Stamp Problem;167
13.3;3 Proof of Theorem 1.1;170
13.4;4 Proof of Theorem 1.2;172
13.5;5 Discussion;179
13.6;References;179
14;A Universal Stein-Tomas Restriction Estimate for Measures in Three Dimensions;181
14.1;1 Introduction;181
14.2;2 Reduction to the Key Geometric Estimate;183
14.3;3 Proof of Theorem 1 and Corollary 1;184
14.4;4 Geometric Estimates: Proof of Corollary 2;185
14.5;References;188
15;On the Exact Order of Asymptotic Bases and Bases for Finite Cyclic Groups;189
15.1;1 Exact Asymptotic Bases;190
15.2;2 Subsets of Exact Asymptotic Bases;190
15.3;3 Exact Order of Asymptotic Bases;192
15.4;4 Postage Stamp Problem;194
15.5;5 Extremal Bases for Finite Cyclic Groups;197
15.6;6 Remarks and Open Problems;198
15.7;References;200
16;The Erdos-Turán Problem in Infinite Groups
;204
16.1;1 The Background;205
16.2;2 The Results;205
16.3;3 The Proofs;206
16.4;References;211
17;A Tiling Problem and the Frobenius Number;212
17.1;1 Introduction;212
17.2;2 Tiling Tori;214
17.3;3 Tiling Rectangles;222
17.3.1;3.1 Cube Tiles;226
17.4;References;229
18;Sumsets and the Convex Hull;230
18.1;1 Introduction;230
18.2;2 A Simplicial Decomposition;232
18.3;3 The Case of a Simplex;233
18.4;4 The General Case;235
18.5;References;236
19;Explicit Constructions of Infinite Families of MSTD Sets;237
19.1;1 Introduction;238
19.2;2 Construction of Infinite Families of MSTD Sets;241
19.3;3 Lower Bounds for the Percentage of MSTDs;243
19.4;4 Concluding Remarks and Future Research;246
19.5;Appendix 1: Size of S(
a, b, c; r);248
19.6;Appendix 2:When Almost ALL Sets are not MSTD Sets
;249
19.7;References;255
20;An Inverse Problem in Number Theory and Geometric Group Theory;257
20.1;1 From Compact Sets to Integers;257
20.2;2 The Inverse Problem;258
20.3;3 Relatively Prime Sets of Lattice Points;263
20.4;Appendix: The Fundamental Observation of Geometric
Group Theory
;264
20.5;References;266
21;Cassels Bases;267
21.1;1 Additive Bases of Finite Order;267
21.2;2 A Lower Bound for Bases of Finite Order;269
21.3;3 Raikov-Stöhr Bases;270
21.4;4 Construction of Thin g-adic Bases of Order h;272
21.5;5 Asymptotically Polynomial Bases;276
21.6;6 Bases of Order 2;278
21.7;7 Bases of Order h 3;285
21.8;8 Notes;292
21.9;References;292
22;Asymptotics of Weighted Lattice Point Counts Inside Dilating Polygons;294
22.1;1 Introduction;294
22.2;2 The Algebraic Case;295
22.3;3 An Almost Everywhere Result;304
22.4;4 Concluding Remarks;307
22.5;References;307
23;Support Bases of Solutions of a Functional Equation Arising From Multiplication of Quantum Integers and the Twin Primes Conjecture
;309
23.1;1 Introduction;309
23.2;2 Main Results;314
23.3;3 Proof of Main Results;314
23.4;References;322
24;Exponential Sums and Distinct Points on Arcs;324
24.1;1 Introduction;324
24.2;2 Three Theorems;325
24.3;3 Two Proofs of Freiman's Lemma;327
24.3.1;3.1 First Proof;327
24.3.2;3.2 Second Proof;328
24.4;4 Proof of Theorem 3;329
24.4.1;4.1 Notation;329
24.4.2;4.2 Arcs;329
24.4.3;4.3 Assumptions;329
24.4.4;4.4 Geometric Progressions;330
24.4.5;4.5 Compactness/Continuity;330
24.4.6;4.6 Dispersion;330
24.4.7;4.7 Perturbation;330
24.4.7.1;4.7.1 Case I;331
24.4.7.2;4.7.2 Case II;332
24.4.8;4.8 Primary Points;332
24.4.9;4.9 Secondary Points;333
24.5;5 Closing Remarks;333
24.6;References;335
25;New Vacca-Type Rational Series for Euler's Constant and Its ''Alternating'' Analog ln4
;336
25.1;1 Introduction;336
25.2;2 Proofs;339
25.3;3 Open Problems;343
25.4;References;344
26;Mixed Sums of Primes and Other Terms;346
26.1;1 Introduction;346
26.2;2 Proofs of Theorems 1.3 and 1.9;351
26.3;3 Discussion of Conjecture 1.5 and Its Variants;354
26.4;References;357
27;Classes of Permutation Polynomials Based on Cyclotomy and an Additive Analogue;359
27.1;1 Introduction;359
27.2;2 Permutation Polynomials from Cyclotomy;360
27.3;3 Permutation Polynomials from Additive Cyclotomy;361
27.4;References;364