Cover -- Title Page -- Contents -- Preface -- Part I Lattice Points and Number Theory -- 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 -- References -- 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 -- References -- 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 -- References -- 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 TwoSquares -- 4.5 A Formula for R(n) -- Problem Set for Section 4.5 -- References -- Part II An Introduction to the Geometry of Numbers -- 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 -- References -- 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 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 -- References -- 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 -- Problem Set for Section 7.3 -- 7.4 Visible Points -- 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 Formsin More Than Two Variables -- 8.5 Approximating by Rational Numbers -- 8.6 Sums of Four Squares -- References -- 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 -- 10 A Minkowski Theorem(Optional) -- 10.1 A Brief History of the Question -- 10.2 A Proof of Minkowski's Theorem -- 10.3 An Application of Minkowski's Theorem -- 10.4 Proving the General Theorem -- References -- Appendix I Gaussian Integers by Peter D. Lax -- I.1 Complex Numbers -- Problem Set for Section I.1 -- I.2 Factorization of Gaussian Integers -- Problem Set for Section I.2 -- I.3. The Fundamental Theorem of Arithmetic -- Problem for Section I.3 -- I.4. Unique Factorization of Gaussian Integers -- Problem for Section I.4 -- I.5. The Gaussian Primes -- I.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 R^2 -- II.3 The Packing of Spheres in Rn -- References -- Appendix III Brief Biographies -- Hermann Minkowski (1864--1909) -- Hans Frederik Blichfeldt (1873{1945) -- References -- Solutions and Hints -- Chapter 1 -- Section 2.1 -- Section 2.2.

Section 2.3 -- Section 3.2 -- Section 3.3 -- Section 4.3 -- Section 4.5 -- Section 5.1 -- Section 5.2 -- Section 5.3 -- Section 6.2 -- Section 6.3 -- Section 7.1 -- Section 7.3 -- Appendix I, Section I.1 -- Appendix I, Section I.2 -- Bibliography -- Index -- About the Authors.