The Geometry of Numbers, Volume 41Cambridge University Press, 22 févr. 2001 - 174 pages 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. |
Table des matières
Lattice Points and Straight Lines | 3 |
Counting Lattice Points | 25 |
Lattice Points and the Area of Polygons | 37 |
Lattice Points in Circles | 47 |
Minkowskis Fundamental Theorem | 65 |
Applications of Minkowskis Theorems | 77 |
Linear Transformations and Integral Lattices | 89 |
Geometric Interpretations of Quadratic Forms | 103 |
A New Principle in the Geometry of Numbers | 117 |
A Minkowski Theorem Optional | 129 |
Gaussian Integers by Peter D | 139 |
The Closest Packing of Convex Bodies | 151 |
Brief Biographies | 157 |
Expressions et termes fréquents
apply approximation Blichfeldt bound boundary called Chapter circle Clearly common complete consider contains convex corresponding covering defined determinant dimensions distance divides divisible draw equal equation examine example exist expressed fact factor Figure formula fractions Gaussian integers geometry of numbers given gives greater Hence inequality infinitely inside interesting interior irrational lattice points least Lemma length lies linear transformation M-set Mathematics means Minkowski's Minkowski's Fundamental Theorem multiplicative number of lattice obtain origin P₁ packing pair parallel parallelogram passes path plane polygon positive positive integer possible prime Problem proof proved quadratic form question rational reader rectangle References relatively prime representations represented sides simple slope solutions sphere squares Suppose symmetry Table Theory translated triangle unit vertices volume write York zero