The Geometry of Numbers

The Geometry of Numbers

C. D. Olds
Anneli Lax
Giuliana P. Davidoff
Volume: 41
Copyright Date: 2000
Edition: 1
Pages: 193
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  • Book Info
    The Geometry of Numbers
    Book Description:

    The Geometry of Numbers 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.

    eISBN: 978-0-88385-955-1
    Subjects: Mathematics
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Table of Contents

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  1. Front Matter (pp. i-viii)
  2. Table of Contents (pp. ix-xii)
  3. Preface (pp. xiii-xvi)
    Giuliana P. Davidoff
  4. Part I. Lattice Points and Number Theory
    • 1 Lattice Points and Straight Lines (pp. 3-24)

      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: thefundamental lattice L...

    • 2 Counting Lattice Points (pp. 25-36)

      Frequently, we will wonder how many lattice points occur on line segments, or inside rectangles, or in various parts of conic sections, and so on. Basically, what we want to know is:How do we count lattice points, or at least estimate their number?This chapter offers some ideas.

      We shall again make use of the arithmetical function [x], defined for every real number x asthe largest integer not exceeding x:

      [x] = largest integer < x .

      We call this integer theintegral part of x. For example, the integral part of 3.6, denoted by [3,6], is 3,...

    • 3 Lattice Points and the Area of Polygons (pp. 37-46)

      Many interesting relationships exist between lattice points and the areas of geometrical figures such as polygons and rectangles. Later, in Part II, we will be exploring Minkowski’s beautiful theorems on this fascinatinggeometry of numbers. This chapter introduces the basic concepts underlying these relationships. We will begin by defining key terms, then examine two important theorems.

      By apolygon, we mean a set of points calledverticesconnected in a given order by line segments calledsides; see Figure 3 .1. To construct a polygon, we number the given points P1, P2, ..., Pn, the draw segments calledsides;see...

    • 4 Lattice Points in Circles (pp. 47-62)

      Among the earliest explorations of lattice points were those undertaken by C. F. Gauss. In 1837 Gauss [3] published a result addressing the question of how many lattice points occur within or on a circle of a certain size. Using our terminology, we would phrase his question this way:

      What is the number N (n) of lattice points in the interior and on the boundary of a circle$C\left( {\sqrt n } \right)$that has radius$r = \sqrt n $and is centered at the origin of the fundamental point-lattice A, with n a nonnegative integer?

      Gauss calculated the numerical results from 10 to 300 presented in...

  5. Part II. An Introduction to the Geometry of Numbers
    • 5 Minkowski’s Fundamental Theorem (pp. 65-76)

      As we said in Section 1.1, thegeometry of numbersis an important branch of number theory that originated in the work of Hermann Minkowski. The reader will find a biographical sketch of this great mathematician in Appendix III.

      The geometry of numbers is connected with the problem of determining whether inequalities of various kinds are solvable in integers. Inequalities for which integer solutions are sought are calledDiophantine inequalities. Earlier, using algebraic methods, Charles Hermite (1822 1901) had proved many general theorems on the solutions of Diophantine inequalities, the most important of which he communicated to Karl Jacobi in...

    • 6 Applications of Minkowski’s Theorems (pp. 77-88)

      For further information on Hermann Minkowski’s discoveries in the geometry of numbers, the interested reader of German should go to his selected papers [4]. There one will find how Minkowski delved deeply into this subject, investigating questions and proving theorems in three dimensions and higher. This chapter will explore some of the ways in which his results help us establish the accuracy possible in approximating real numbers by rational numbers.

      As our first application of Minkowski’s Fundamental Theorem, we shall prove the following approximation. It is illustrated by the parallelogram shown in Figure 6.1.

      Theorem 6.1. Given any real number...

    • 7 Linear Transformations and Integral Lattices (pp. 89-102)

      Those of you who read the optional Section 5. 4 are probably already familiar with much of the material we are about to discuss. For those readers who are encountering the geometry of numbers for the first time, we hope that you will be stimulated to understand the proofs of more difficult theorems or even go on to do independent study in the literature. A prerequisite for more advanced work of this sort is some knowledge oflinear transformations. In this chapter, we will get a sense of how linear transformations apply to the study of integral lattices. We begin...

    • 8 Geometric Interpretations of Quadratic Forms (pp. 103-118)

      The study ofquadratic forms in two or more variablestakes us into some of the most advanced parts of the theory of numbers. The whole third volume of Dickson’sHistory of the Theory of Numbers[6 ], for example, is devoted to the subject—and Dickson stopped with the 1920s! It was Joseph Louis Lagrange (1736–1813), the foremost mathematician of the eighteenth century (rivalled only by Euler), who published the first proof that every positive integer can be expressed as the sum of at most four squares (see Section 8.6). Lagrange’s theory of quadratic forms, first developed in...

    • 9 A New Principle in the Geometry of Numbers (pp. 119-128)

      Around 1891, Hermann Minkowski discovered his Fundamental Theorem, opening up a new field of study which he called thegeometry of numbers. Using his theorem and its generalizations, Minkowski was able to solve many difficult problems in number theory. In Chapter 6 we examined some of the easier applications of Minkowski’s theorems.

      Despite the excitement aroused by Minkowski’s groundbreaking work, it was another 15 years before any new principle in the geometry of numbers was discovered. The credit for this breakthrough goes to Hans Frederik Blichfeldt, who in 1914 published a theorem from which a great portion of the geometry...

    • 10 A Minkowski Theorem (Optional) (pp. 129-138)

      In 1866, Pafnuty Livovich Tchebychev (1821–1894), one of Russia’s greatest mathematicians, proved the following theorem.

      Theorem 10.1.Let θ be irrational and suppose that α is any real number for which the equation x − θy − α = 0 has no solutions in integers p, q. Then for any given positive ε, there are infinitely many pairs of integers p, q such that

      |q(p − θq − α)| < 2, (10.1)

      and, at the same time, such that |p − θq − α| < ε.

      Tchebychev’s proof was long, running to forty pages [7], and was purely arithmetical, making...

  6. Appendix I Gaussian Integers (pp. 139-150)
    Peter D. Lax
  7. Appendix II The Closest Packing of Convex Bodies (pp. 151-156)
  8. Appendix III Brief Biographies (pp. 157-160)
  9. Solutions and Hints (pp. 161-168)
  10. Bibliography (pp. 169-171)
  11. Index (pp. 172-174)
  12. About the Authors (pp. 175-176)


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