Approximation by Algebraic Numbers (Cambridge Tracts in Mathematics)
Cambridge University Press, 11/8/2004
EAN 9780521823296, ISBN10: 0521823293
Hardcover, 292 pages, 22.9 x 15.2 x 2.1 cm
Language: English
Algebraic numbers can approximate and classify any real number. Here, the author gathers together results about such approximations and classifications. Written for a broad audience, the book is accessible and self-contained, with complete and detailed proofs. Starting from continued fractions and Khintchine's theorem, Bugeaud introduces a variety of techniques, ranging from explicit constructions to metric number theory, including the theory of Hausdorff dimension. So armed, the reader is led to such celebrated advanced results as the proof of Mahler's conjecture on S-numbers, the Jarnik–Besicovitch theorem, and the existence of T-numbers. Brief consideration is given both to the p-adic and the formal power series cases. Thus the book can be used for graduate courses on Diophantine approximation (some 40 exercises are supplied), or as an introduction for non-experts. Specialists will appreciate the collection of over 50 open problems and the rich and comprehensive list of more than 600 references.
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.
'The book is written in a relaxed style, and begins with some accessible introductory chapters ... It is nicely written and well explained, and proofs in the main are given in full. this book is certainly suitable for a non-expert in the area, or as a graduate course for an advanced student ... All in all, this is a very nice book.' Bulletin of the London Mathematical Society