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-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 361 Seiten

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