Résumé
This is a self-contained introduction to the geometry of
numbers, beginning with easily understood questions about
lattice points on lines, circles and inside simple polygons
in the plane. A minimum of mathematical expertise is
required beyond an acquaintance with elementary geometry.
The authors gradually lead up to the theorems of Minkowski
and others who succeeded him. On the way the reader will
see how this powerful approach gives improved
approximations to irrational numbers by rationals,
simplifies arguments on ways of representing integers as
sums of squares, and provides a natural tool for attacking
problems involving dense packings of spheres.
Contents
- Preface xiii
- Chapter 1 Lattice Points and Straight Lines
-
- 1.1 The Fundamental Lattice 3
- 1.2 Lines in Lattice Systems 4
- 1.3 Lines with Rational Slope 6
- 1.4 Lines with Irrational Slope 11
- 1.5 Broadest Paths without Lattice Points 18
- 1.6 Rectangles on Paths without Lattice Points 20
- Problem Set for Chapter 1 23
- Chapter 2 Counting Lattice Points
-
- 2.1 The Greatest Integer Function, [x] 25
- Problem Set for Section 2.1 27
- 2.2 Positive Integral Solutions of ax + by = n 28
- Problem Set for Section 2.2 31
- 2.3 Lattice Points inside a Triangle 32
- Problem Set for Section 2.3 34
- Chapter 3 Lattice Points and the Area of Polygons
-
- 3.1 Points and Polygons 37
- 3.2 Pick's Theorem 38
- Problem Set for Section 3.2 39
- 3.3 A Lattice Point Covering Theorem for Rectangles 40
- Problem Set for Section 3.3 45
- Chapter 4 Lattice Points in Circles
-
- 4.1 How Many Lattice Points Are There? 47
- 4.2 Sums of Two Squares 50
- 4.3 Numbers Representable as a Sum of Two Squares 53
- Problem Set for Section 4.3 56
- 4.4 Representations of Prime Numbers as Sums of Two Squares 56
- 4.5 A Formula for R(n) 58
- Problem Set for Section 4.5 60
- Chapter 5 Minkowski's Fundamental Theorem
-
- 5.1 Minkowski's Geometric Approach 65
- Problem Set for Section 5.1 66
- 5.2 Minkowski M-Sets 67
- Problem Set for Section 5.2 69
- 5.3 Minkowski's Fundamental Theorem 69
- Problem Set for Section 5.3 74
- 5.4 (Optional) Minkowski's Theorem in n Dimensions 75
- Chapter 6 Applications of Minkowski's Theorems
-
- 6.1 Approximating Real Numbers 77
- 6.2 Minkowski's First Theorem 78
- Problem Set for Section 6.2 81
- 6.3 Minkowski's Second Theorem 81
- Problem for Section 6.3 82
- 6.4 Approximating Irrational Numbers 82
- 6.5 Minkowski's Third Theorem 84
- 6.6 Simultaneous Diophantine Approximations 85
- Reading Assignment for Chapter 6 86
- Chapter 7 Linear Transformations and Integral Lattices
-
- 7.1 Linear Transformations 89
- Problem Set for Section 7.1 92
- 7.2 The General Lattice 92
- 7.3 Properties of the Fundamental Lattice A 94
- Problem Set for Section 7.3 99
- 7.4 Visible Points 99
- Chapter 8 Geometric Interpretations of Quadratic Forms
- 8.1 Quadratic Representation 103
- 8.2 An Upper Bound for the Minimum Positive Value 104
- 8.3 An Improved Upper Bound 107
- 8.4 (Optional) Bounds for the Minima of Quadratic Forms in More Than Two Variables 110
- 8.5 Approximating by Rational Numbers 111
- 8.6 Sums of Four Squares 113
- Chapter 9 A New Principle in the Geometry of Numbers
- 9.1 Blichfeldt's Theorem 119
- 9.2 Proof of Blichfeldt's Theorem 120
- 9.3 A Generalization of Blichfeldt's Theorem 121
- 9.4 A Return to Minkowski's Theorem 123
- 9.5 Applications of Blichfeldt's Theorem 125
- Chapter 10 A Minkowski Theorem (Optional)
- 10.1 A Brief History of the Question 129
- 10.2 A Proof of Minkowski's Theorem 130
- 10.3 An Application of Minkowski's Theorem 135
- 10.4 Proving the General Theorem 137
- Appendix I Gaussian Integers, by Peter D. Lax
- I.1 Complex Numbers 139
- Problem Set for Section I.1 140
- I.2 Factorization of Gaussian Integers 140
- Problem Set for Section I.2 141
- I.3 The Fundamental Theorem of Arithmetic 141
- Problem for Section I.3 144
- I.4 Unique Factorization of Gaussian Integers 144
- Problem for Section I.4 145
- I.5 The Gaussian Primes 145
- I.6 More about Gaussian Primes 148
- Appendix II The Closest Packing of Convex Bodies
- II.1 Lattice-Point Packing 151
- II.2 Closest Packing of Circles in R[superscript 2] 152
- II.3 The Packing of Spheres in R[superscript n] 153
- Appendix III Brief Biographies
- Hermann Minkowski 157
- Hans Frederik Blichfeldt 159
- Solutions and Hints 161
- Bibliography 169
- Index
Caractéristiques techniques
| PAPIER | |
| Éditeur(s) | Cambridge University Press |
| Auteur(s) | C. D. Olds, Anneli Lax, Giuliana Davidoff |
| Parution | 16/10/2001 |
| Nb. de pages | 176 |
| Format | 15,3 x 22,8 |
| Couverture | Broché |
| Poids | 274g |
| Intérieur | Noir et Blanc |
| EAN13 | 9780883856437 |
| ISBN13 | 978-0-88385-643-7 |
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