Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Symmetry in the sequence of approximation coefficients

Author: Avraham Bourla
Journal: Proc. Amer. Math. Soc. 141 (2013), 3681-3688
MSC (2010): Primary 11A55, 11J70, 11J71, 11J81; Secondary 01-02
Published electronically: July 17, 2013
MathSciNet review: 3091760
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{a_n\}_1^\infty $ and $ \{\theta _n\}_0^\infty $ be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function $ f$ such that $ a_{n+1} = f(\theta _{n\pm 1},\theta _n)$. In tandem with a formula due to Dajani and Kraaikamp, we will write $ \theta _{n \pm 1}$ as a function of $ (\theta _{n \mp 1}, \theta _n)$, revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.

References [Enhancements On Off] (What's this?)

  • 1. F. Bagemihl and J. R. McLaughlin, Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions, J. Reine Angew. Math. 221 (1966), 146–149. MR 0183999
  • 2. Edward B. Burger, Exploring the number jungle: a journey into Diophantine analysis, Student Mathematical Library, vol. 8, American Mathematical Society, Providence, RI, 2000. MR 1774066
  • 3. Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, Mathematical Surveys and Monographs, vol. 30, American Mathematical Society, Providence, RI, 1989. MR 1010419
  • 4. Karma Dajani and Cor Kraaikamp, Ergodic theory of numbers, Carus Mathematical Monographs, vol. 29, Mathematical Association of America, Washington, DC, 2002. MR 1917322
  • 5. W. B. Jurkat and A. Peyerimhoff, Characteristic approximation properties of quadratic irrationals, Internat. J. Math. & Math. Sci.(1), 1978. MR 0517950 (80a:10050)
  • 6. O. Perron, $ \ddot {U}$ber die approximation irrationaler zahlen durch rationale, Heindelberg Akad. Wiss. Abh.(4), 1921.
  • 7. J. Tong, The conjugate property of the Borel theorem on diophantine approximation, Math. Z. 184(2), 1983. MR 0716268 (85m:11039)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11A55, 11J70, 11J71, 11J81, 01-02

Retrieve articles in all journals with MSC (2010): 11A55, 11J70, 11J71, 11J81, 01-02

Additional Information

Avraham Bourla
Affiliation: Department of Mathematics. Saint Mary’s College of Maryland, Saint Mary’s City, Maryland 20686
Address at time of publication: Department of Mathematics and Statistics, American University, 4400 Massachusetts Avenue, NW, Washington, DC 20016

Received by editor(s): October 16, 2011
Received by editor(s) in revised form: December 1, 2011, and December 8, 2011
Published electronically: July 17, 2013
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society