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The Geometry of Numbers AMS / MAA ANNELI LAX NEW MATHEMATICAL LIBRARY VOL 41 The Geometry of Numbers C.D.Olds Anneli Lax Giuliana Davidoff 10.1090/nml/041 The Geometry of Numbers ©Copyright 2000 by The Mathematical Association of America (Inc.) All rights reserved under International and Pan-American Copyright Conventions. Published in Washington, D.C. by The Mathematical Association of America Library of Congress Catalog Card Number: 00-110657 Print ISBN 978-0-88385-643-7 Electronic ISBN 978-0-88385-955-1 Manufactured in the United States of America The Geometry of Numbers C. D. Olds Anneli Lax Giuliana P. Davidoff Published and Distributed by The Mathematical Association of America Dedicated to Dr. Anneli Lax Editor of the New Mathematical Library from 1961 to 1991 The Geometry of Numbers is one of the last editorial projects Dr. Lax worked on before her death in October 1999. She observed that we cannot know what Dr. Olds would have added to his book had he lived. Neither do we know how she herself would have organized a final version of the text. I hope that this finished work does justice to the effort and vision of each of them. –G.P.D. ANNELI LAX NEW MATHEMATICAL LIBRARY PUBLISHED BY THE MATHEMATICAL ASSOCIATION OF AMERICA Editorial Committee Michael J. McAsey, Editor Arthur Benjamin Lester H. Lange George Berzsenyi Mark E. Saul Richard K. Guy Philip D. Straffin Ross A. Honsberger Joan Wyzkoski Weiss The New Mathematical Library (NML) was started in 1961 by the School Mathematics Study Group to make available to high school students short expository books on various topics not usually covered in the high school syllabus. In a decade the NML matured into a steadily growing series of some twenty titles of interest not only to the originally intended audience, but to college students and teachers at all levels. Previously published by Random House and L. W. Singer, the NML became a publication series of the Mathematical Association of America (MAA) in 1975. Under the auspices of the MAA the NML continues to grow and remains dedicated to its original and expanded purposes. In its third decade, it contains forty titles. ANNELI LAX NEW MATHEMATICAL LIBRARY 1. Numbers: Rational and Irrational by Ivan Niven 2. What is Calculus About? by W. W. Sawyer 3. An Introduction to Inequalities by E. F. Beckenbach and R. Bellman 4. Geometric Inequalities by N. D. Kazarinoff 5. The Contest Problem Book I Annual High School Mathematics Examinations 1950{1960. Compiled and with solutions by Charles T. Salkind 6. The Lore of Large Numbers by P. J. Davis 7. Uses of Infinity by Leo Zippin 8. Geometric Transformations I by I. M. Yaglom, translated by A. Shields 9. Continued Fractions by Carl D. Olds 10. Replaced by NML-34 11. Hungarian Problem Books I and II, Based on the Eotv¨ os¨ Competitions 12. ff 1894{1905 and 1906{1928, translated by E. Rapaport 13. Episodes from the Early History of Mathematics by A. Aaboe 14. Groups and Their Graphs by E. Grossman and W. Magnus 15. The Mathematics of Choice by Ivan Niven 16. From Pythagoras to Einstein by K. O. Friedrichs 17. The Contest Problem Book II Annual High School Mathematics Examinations 1961{1965. Compiled and with solutions by Charles T. Salkind 18. First Concepts of Topology by W. G. Chinn and N. E. Steenrod 19. Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer 20. Invitation to Number Theory by Oystein Ore 21. Geometric Transformations II by I. M. Yaglom, translated by A. Shields 22. Elementary Cryptanalysis–A Mathematical Approach by A. Sinkov 23. Ingenuity in Mathematics by Ross Honsberger 24. Geometric Transformations III by I. M. Yaglom, translated by A. Shenitzer 25. The Contest Problem Book III Annual High School Mathematics Examinations 1966{1972. Compiled and with solutions by C. T. Salkind and J. M. Earl 26. Mathematical Methods in Science by George Polya 27. International Mathematical Olympiads–1959œ1977. Compiled and with solutions by S. L. Greitzer 28. The Mathematics of Games and Gambling by Edward W. Packel 29. The Contest Problem Book IV Annual High School Mathematics Examinations 1973{1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione, and N. Shell 30. The Role of Mathematics in Science by M. M. Schiffer and L. Bowden 31. International Mathematical Olympiads 1978{1985 and forty supplementary problems. Compiled and with solutions by Murray S. Klamkin 32. Riddles of the Sphinx by Martin Gardner 33. U.S.A. Mathematical Olympiads 1972{1986. Compiled and with solutions by Murray S. Klamkin 34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J. Wilson 35. Exploring Mathematics with Your Computer by Arthur Engel 36. Game Theory and Strategy by Philip D. Straffin, Jr. 37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross Honsberger 38. The Contest Problem Book V American High School Mathematics Examinations and American Invitational Mathematics Examinations 1983{1988. Compiled and augmented by George Berzsenyi and Stephen B. Maurer 39. Over and Over Again by Gengzhe Chang and Thomas W. Sederberg 40. The Contest Problem Book VI American High School Mathematics Examinations 1989{1994. Compiled and augmented by Leo J. Schneider 41. The Geometry of Numbers by C.D. Olds, Anneli Lax, and Giuliana Davidoff Other titles in preparation. Books may be ordered from: MAA Service Center P. O. Box 91112 Washington, DC 20090-1112 1-800-331-1622 fax: 301-206-9789 Contents Preface ................................................... ...... xiii Part I. Lattice Points and Number Theory 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 ix x The Geometry of Numbers 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 Part II. An Introduction to the Geometry of Numbers 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 Λ ................... 94 Problem Set for Section 7.3............................... 99 7.4 Visible Points............................................99 Contents xi 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
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