Book contents
- Frontmatter
- Contents
- Preface
- Frequently used notation
- 1 Distribution modulo one
- 2 On the fractional parts of powers of real numbers
- 3 On the fractional parts of powers of algebraic numbers
- 4 Normal numbers
- 5 Further explicit constructions of normal and non-normal numbers
- 6 Normality to different bases
- 7 Diophantine approximation and digital properties
- 8 Digital expansion of algebraic numbers
- 9 Continued fraction expansions and β-expansions
- 10 Conjectures and open questions
- Appendix A Combinatorics on words
- Appendix B Some elementary lemmata
- Appendix C Measure theory
- Appendix D Continued fractions
- Appendix E Diophantine approximation
- Appendix F Recurrence sequences
- References
- Index
Preface
Published online by Cambridge University Press: 05 October 2012
- Frontmatter
- Contents
- Preface
- Frequently used notation
- 1 Distribution modulo one
- 2 On the fractional parts of powers of real numbers
- 3 On the fractional parts of powers of algebraic numbers
- 4 Normal numbers
- 5 Further explicit constructions of normal and non-normal numbers
- 6 Normality to different bases
- 7 Diophantine approximation and digital properties
- 8 Digital expansion of algebraic numbers
- 9 Continued fraction expansions and β-expansions
- 10 Conjectures and open questions
- Appendix A Combinatorics on words
- Appendix B Some elementary lemmata
- Appendix C Measure theory
- Appendix D Continued fractions
- Appendix E Diophantine approximation
- Appendix F Recurrence sequences
- References
- Index
Summary
primitive Mathematik
hohe Kunst
Thomas BernhardUn chercheur universitaire
est un individu qui en sait toujours plus
sur un sujet toujours moindre,
en sorte qu'il finit par savoir tout de rien.
Simon LeysThree of the main questions that motivate the present book are the following:
Is there a transcendental real number α such that ∥αn∥ tends to 0 as n tends to infinity?
Is the sequence of fractional parts {(3/2)n}, n ≥ 1, dense in the unit interval?
What can be said on the digital expansion of an irrational algebraic number?
The latter question amounts to the study of the sequence (ξ10n)n≥1 modulo one, where ξ is an irrational algebraic number. More generally, for given real numbers ξ ≠ 0 and α > 1, we are interested in the distribution of the sequences ({ξαn})n≥1 and (∥ξαn∥)n≥1, where {·} (resp., ∥·∥) denotes the fractional part (resp., the distance to the nearest integer). The situation is very well understood from a metrical point of view. However, for a given pair (ξ, α), our knowledge on ({ξαn})n≥1 is extremely limited, except in very few cases. For instance when ξ = 1 and α is a Pisot number, that is, an algebraic integer (an algebraic integer is an algebraic number whose minimal defining polynomial over ℤ is monic) all of whose Galois conjugates (except itself) are lying in the open unit disc, it is not difficult to show that ∥αn∥ tends to 0 as n tends to infinity.
- Type
- Chapter
- Information
- Distribution Modulo One and Diophantine Approximation , pp. ix - xivPublisher: Cambridge University PressPrint publication year: 2012