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
1 - Approximation by rational 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
Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ≥1 → : ℝ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q| < Ψ (q) has infinitely many solutions in rational numbers p/q? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions.
Dirichlet and Liouville
Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b, there exists an integer a with |ξ – a/b| ≤ 1/(2b), and one may hope that there are infinitely many integers b for which |ξ – a/b| is in fact much smaller than 1/(2b). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.
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- Approximation by Algebraic Numbers , pp. 1 - 26Publisher: Cambridge University PressPrint publication year: 2004