The Geometry of Numbers

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Brand new. We distribute directly for the publisher. The geometry of numbers originated with the publication of Minkowski's seminal work in 1896 and ultimately established itself ... as an important field in its own right. By resetting various problems into geometric contexts, it sometimes allows difficult questions in arithmetic or other areas of mathematics to be answered more easily; inevitably, it lends a larger, richer perspective to the topic under investigation. Its principal focus is the study of lattice points, or points in n-dimensional space with integer coordinates-a subject with an abundance of interesting problems and important applications. Advances in the theory have proved highly significant for modern science and technology, yielding new developments in crystallography, superstring theory, and the design of error-detecting and error-correcting codes by which information is stored, compressed for transmission, and received. This book presents 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. Little mathematical expertise is required beyond an acquaintance with those objects and with some basic results in geometry. The reader moves gradually to theorems of Minkowski and others who succeeded him. On the way, he or she 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. An appendix by Peter Lax gives a lovely geometric proof of the fact that the Gaussian integers form a Euclidean domain, characterizing the Gaussian primes, and proving that unique factorization holds there. In the process, he provides yet another glimpse into the power of a geometric approach to number theoretic problems. Read more Show Less

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Overview

This book presents 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. Little mathematical expertise is required beyond an acquaintance with those objects and with some basic results in geometry.

The reader moves gradually to theorems of Minkowski and others who succeeded him. On the way, he or she 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.

An appendix by Peter Lax gives a lovely geometric proof of the fact that the Gaussian integers form a Euclidean domain, characterizing the Gaussian primes, and proving that unique factorization holds there. In the process, he provides yet another glimpse into the power of a geometric approach to number theoretic problems.

The geometry of numbers originated with the publication of Minkowski's seminal work in 1896 and ultimately established itself as an important field in its own right. By resetting various problems into geometric contexts, it sometimes allows difficult questions in arithmetic or other areas of mathematics to be answered more easily; inevitably, it lends a larger, richer perspective to the topic under investigation. Its principal focus is the study of lattice points, or points in n-dimensional space with integer coordinates-a subject with an abundance of interesting problems and important applications. Advances in the theory have proved highly significant for modern science and technology, yielding new developments in crystallography, superstring theory, and the design of error-detecting and error-correcting codes by which information is stored, compressed for transmission, and received.

Read More Show Less

Editorial Reviews

Gary MacGillivray
"The Geometry of Numbers is a very well-written expository book. It is well paced and enjoyable to read. The material is interesting, well-chosen, and presented a level appropriate for high-school students, undergraduates and teachers at all levels."
crux Mathematicorum
Jorg M. Willis
"Minkowski's ingenious idea, the interplay between number theory and geometry, can be pursued throughout the booklet. So it is a good appetizer for students."
Zentralblatt MATH
Zentrablatt fur Mathematik
"[This book] provides yet another glimpse into the power of a geometric approach to number theoretic problems."
Booknews
Presents 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, then focusing on the major theorems of Minkowski and Blichfeldt. Suitable for interested high school students and nonprofessionals, the volume presents solutions or hints for problems at the back of the book. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780883856437
  • Publisher: Mathematical Association of America
  • Publication date: 2/28/2001
  • Series: New Mathematical Library Series
  • Edition description: New Edition
  • Pages: 168
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.83 (d)

Table of Contents

Preface
Part I. Lattice Points and Number Theory
Chapter 1 Lattice Points and Straight Lines
1.1 The Fundamental Lattice
1.2 Lines in Lattice Systems
1.3 Lines with Rational Slope
1.4 Lines with Irrational Slope
1.5 Broadest paths without Lattice Points
1.6 Rectangles on Paths without Lattice Points
Problem Set for Chapter 1

Chapter 2 Counting Lattice Points
2.1 The Greatest Integer Function, [x]
 Problem Set for Section 2.1
2.2 Positive Integral Solutions of ax+by=n
 Problem Set for Section 2.2
2.3 Lattice Points inside a Triangle
 Problem set for Section 2.3

Chapter 3 Lattice Points and the Area of Polygons
3.1 Points and Polygons
3.2 Pick's Theorem
 Problem Set for Section 3.2
3.3 A Lattice Point Covering Theorem for Rectangles
 Problem Set for Section 3.3

Chapter 4 Lattice Points in Circles
4.1 How Many Lattice Points Are There?
4.2 Sums of Two Squares
4.3 Numbers Representable as a Sum of Two Squares
 Problem Set for Section 4.3
4.4 Representations of Prime Numbers as Sums of Two Squares
4.5 A Formula for R(n)
 Problem Set for Section 4.5

Part II An introduction of the Geometry of Numbers

Chapter 5 Minkowski's Fundamental Theorem
5.1 Minkowski's Geometric Approach
 Problem Set for Section 5.1
5.2 Minkowski M-Sets
 Problem Set for Section 5.2
5.3 Minkowski's Fundamental Theorem
 Problem Set for Section 5.3
5.4 (Optional) Minkowski's Theorem in n Dimensions

Chapter 6 Applications of Minkowski's Theorems
6.1 Approximating Real Numbers
6.2 Minkowski's First Theorem
 Problem Set for Section 6.2
6.3 Minkowski's Second Theorem
 Problem Set for Section 6.3
6.4 Approximating Irrational Numbers
6.5 Minkowski's Third Theorem
6.6 Simultaneous Diophantine Approximations
Reading Assignment for Chapter 6

Chapter 7 Linear Transformations and Integral Lattices
7.1 Linear Transformations
 Problem Set for Section 7.1
7.2 The General Lattice
7.3 Properties of the Fundamental lattice A
 Problem Set for Section 7.3
7.4 Visible Points

Chapter 8 Geometric Interpretations of Quadratic Forms
8.1 Quadratic Representation
8.2 An Upper Bound for the Minimum Positive Value
8.3 An Improved Upper Bound
8.4 (Optional) Bounds for the Minima of Quadratic Forms in More Than Two Variables
8.5 Approximating by Rational Numbers
8.6 Sums of Four Squares

Chapter 9 A New Principle in the Geometry of Numbers
9.1 Blichfeldt's Theorem
9.2 Proof of Blichfeldt's Theorem
9.3 A Generalization of Blichfeldt's Theorem
9.4 A Return to Minkowski's Theorem
9.5 Applications of Blichfeldt's Theorem

Chapter 10 A Minkowski Theorem (Optional)
10.1 A Brief History of the Questions
10.2 A Proof of Minkowski's Theorem
10.3 An Application of Minkowski's Theorem
10.4 Proving the General Theorem

Appendix I Gaussian Integers by Peter D. Lax
1.1 Complex Numbers
Problem Set for Section 1.1
1.2  Factorization of Gaussian Integers
Problem Set for Section 1.2
1.3 The Fundamental Theorem of Arithmetic
Problem Set for Section 1.3
1.4 Unique Factorization of Gaussian Integers
Problem for Section 1.4
1.5 The Gaussian Primes
1.6 More about Gaussian Primes

Appendix II The Closest Packing of Convex Bodies
II.1 Lattice-Point Packing
II.2 Closest Packing of Circles in Rsquared
II.3 The Packings of Spheres in R to the nth Power

Appendix III Brief Biographies
 Hermann Minkowski
 Hans Fredrik Blichfeldt
Solutions and Hints
Bibliography
Index

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First Chapter

1.1 The Fundamental Lattice The theme of this book is the geometry of numbers, a branch of the theory of numbers that was discovered by Hermann Minkowski (1864-1909). Where other mathematicians had attacked problems of certain types algebraically, Minkowski's genius was to approach them from a geometrical point of view. Through the visible order of geometrical constructs, he was able to reveal and explore many numerical relationships. We shall trace Minkowski's explorations in the second part of this book. Here, we begin at the beginning, by defining the two elementary concepts on which the entire geometry of numbers rests: the fundamental lattice Land the fundamental point-lattice A that L determines. When we speak of lattice systems, we are imagining grids of points in space connected like monkey bars on a playground. We can construct a simple planar lattice on an ordinary rectangular coordinate system just by drawing straight lines. First we draw lines parallel to the y-axis through the points ..., ((-2,0), (-1,0), (0,0), (1,0), (2,0), (3,0),..., and then we draw lines parallel to the x-axis through the points ..., (0,-2), (0,-1),(0,0),(0,1), (0,2),..., These lines form the fundamental lattice L. The points where these lines intersect, called lattice points, are the vertices of squares. Lattice points are points with integer coordinates (x,y), as Figure 1.1 indicates. For clarity, to distinguish between a lattice point and any point (x,y), we shall designate a lattice point by (p,q), wher ex=p, y=q are integers. It is the lattice points that determine the fundamental point-lattice A. For now, we shall consider only lattices in the plane, saving n-dimensional figures for Part II. Evenly spaced along lines running parallel to the x- and y- axes, at first glance lattice points seem to be of little interest in themselves. In 1837, Carl Friedrich Gauss (1777-1855) [1] published a seminal paper on the number of lattice points contained within and on a circle of radius r. Since then, mathematicians have proposed a great many intriguing related problems. Many of these problems have now been answered; some are only partially solved; and others-including the optimal form of Gauss's original questions- still defy solutions. To prepare ourselves for examining such problems, we must first discuss some of the relationships between straight lines and lattice points.
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