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
6 - Deeper results on the measure of exceptional sets
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
Theorem 5.4, due to A. Baker and Schmidt [45], asserts that for any integer n ≥ 1 and any real number τ ≥ 1 the Hausdorff dimension of the set W*n (τ) of real numbers ξ with w*n (ξ) = τ(n + 1) – 1 is equal to 1/τ. In the present Chapter, we are concerned with various refinements, including the determination of the Hausdorff measure of W*n (τ) at the critical exponent (Corollary 6.3 below).
There are essentially two new ingredients. On the one hand, we need an improvement of Proposition 5.4, which is due to Beresnevich [61] and asserts that real algebraic numbers of bounded degree are distributed ‘as evenly as they could be’. On the other hand, we present a refined analysis of the Hausdorff measure of sets of real numbers close to infinitely many points in a given real sequence.
One essential tool, introduced in Section 6.1, is the notion of ‘optimal regular systems’ (also termed ‘best possible regular systems’ by Beresnevich, Bernik, and Dodson [67]). We state four general results on sets of real numbers close to infinitely many points in an optimal regular system. We establish the first one in Section 6.2, which allows us to give an alternative proof of (a slightly stronger form of) Khintchine's Theorem 1.10. The second one, stated in Section 6.3, provides the Hausdorff dimension of general exceptional sets.
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- Approximation by Algebraic Numbers , pp. 122 - 138Publisher: Cambridge University PressPrint publication year: 2004