Book contents
- Frontmatter
- Contents
- Preface
- Frequently used notation
- 1 Approximation by rational numbers
- 2 Approximation to algebraic numbers
- 3 The classifications of Mahler and Koksma
- 4 Mahler's Conjecture on S-numbers
- 5 Hausdorff dimension of exceptional sets
- 6 Deeper results on the measure of exceptional sets
- 7 On T-numbers and U-numbers
- 8 Other classifications of real and complex numbers
- 9 Approximation in other fields
- 10 Conjectures and open questions
- Appendix A Lemmas on polynomials
- Appendix B Geometry of numbers
- References
- Index
Appendix B - Geometry of numbers
Published online by Cambridge University Press: 12 August 2009
- Frontmatter
- Contents
- Preface
- Frequently used notation
- 1 Approximation by rational numbers
- 2 Approximation to algebraic numbers
- 3 The classifications of Mahler and Koksma
- 4 Mahler's Conjecture on S-numbers
- 5 Hausdorff dimension of exceptional sets
- 6 Deeper results on the measure of exceptional sets
- 7 On T-numbers and U-numbers
- 8 Other classifications of real and complex numbers
- 9 Approximation in other fields
- 10 Conjectures and open questions
- Appendix A Lemmas on polynomials
- Appendix B Geometry of numbers
- References
- Index
Summary
Geometry of numbers turns out to be a very useful tool in Diophantine approximation. For instance, it allows us to construct non-zero integer polynomials taking small values at prescribed points. In the course of the book, we applied several times the ‘first Theorem of Minkowski’ and the ‘second Theorem of Minkowski’, which are Theorems B.2 and B.3 below, respectively. We give a full proof of Theorem B.2, but not of Theorem B.3, which is much deeper. Throughout this Appendix. n denotes a positive integer. A set C in ℝn having inner points and contained in the closure of its open kernel is called a body (or a domain).
theorem B.1. Let C be a bounded convex body in ℝn, symmetric about the origin and of volume vol(C). If vol(C) > 2nor if vol(C) = 2nand C is compact, then C contains a point with integer coordinates, other than the origin.
proof. This proof is due to Mordell [429]. By classical arguments from elementary topology, it is enough to treat the case where vol(C) > 2n. For any positive integer m, denote by Cm the set of points of C having rational coordinates with denominator m. As m tends to infinity, the cardinality of Cm becomes equivalent to vol(C)mn, and is thus strictly larger than (2m)n when m is large enough.
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- Information
- Approximation by Algebraic Numbers , pp. 235 - 239Publisher: Cambridge University PressPrint publication year: 2004