'The mother of all continued fractions'

Karma Dajani; Cor Kraaikamp

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 1, page 109-123
  • ISSN: 0010-1354

Abstract

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We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full continued fraction), and obtain the ergodic system underlying the Farey expansion.

How to cite

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Dajani, Karma, and Kraaikamp, Cor. "'The mother of all continued fractions'." Colloquium Mathematicae 84/85.1 (2000): 109-123. <http://eudml.org/doc/210790>.

@article{Dajani2000,
abstract = {We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion\}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full continued fraction), and obtain the ergodic system underlying the Farey expansion.},
author = {Dajani, Karma, Kraaikamp, Cor},
journal = {Colloquium Mathematicae},
keywords = {insertion; ergodic theory; continued fractions; singularization; semi-regular continued fractions},
language = {eng},
number = {1},
pages = {109-123},
title = {'The mother of all continued fractions'},
url = {http://eudml.org/doc/210790},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Dajani, Karma
AU - Kraaikamp, Cor
TI - 'The mother of all continued fractions'
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 1
SP - 109
EP - 123
AB - We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full continued fraction), and obtain the ergodic system underlying the Farey expansion.
LA - eng
KW - insertion; ergodic theory; continued fractions; singularization; semi-regular continued fractions
UR - http://eudml.org/doc/210790
ER -

References

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  1. [AF] R. L. Adler and L. Flatto, The backward continued fraction map and geodesic flow, Ergodic Theory Dynam. Systems 4 (1984), 487-492. Zbl0563.58019
  2. [B] W. Bosma, Approximation by mediants, Math. Comp. 54 (1990), 421-434. Zbl0697.10042
  3. [BY] G. Brown and Q. H. Yin, Metrical theory for Farey continued fractions, Osaka J. Math. 33 (1996), 951-970. Zbl0880.11057
  4. [C] A. L. Cauchy, Oeuvres, Gauthier-Villars, Paris, 1890-1895. 
  5. [G] J. R. Goldman, Hurwitz sequences , the Farey process , and general continued fractions, Adv. Math. 72 (1988), 239-260. Zbl0667.10007
  6. [I] S. Ito, Algorithms with mediant convergents and their metrical theory, Osaka J. Math. 26 (1989), 557-578. Zbl0702.11046
  7. [Kh] A. Ya. Khintchine, Metrische Kettenbruchproblem, Compositio Math. 1 (1935), 361-382. 
  8. [K] C. Kraaikamp, A new class of continued fraction expansions, Acta Arith. 57 (1991), 1-39. Zbl0721.11029
  9. [L] J. Lehner, Semiregular continued fractions whose partial denominators are 1 or 2 , in: The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions (Brooklyn, NY, 1992), Contemp. Math. 169, Amer. Math. Soc., Providence, RI, 1994, 407-410. Zbl0814.11008
  10. [R] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 472-493. Zbl0079.08901
  11. [Rich] I. Richards, Continued fractions without tears, Math. Mag. 54 (1981), 163-171. Zbl0466.10003
  12. [Z] D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, Berlin, 1981. 

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