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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11601-1

Published electronically:
July 17, 2013

MathSciNet review:
3091760

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Abstract | References | Similar Articles | Additional Information

Abstract: Let and be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function such that . In tandem with a formula due to Dajani and Kraaikamp, we will write as a function of , revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.

**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. MR**0183999 (32 #1475)****2.**E. B. Burger,*Exploring the number jungle: A journey into diophantine analysis*, Providence, RI: Amer. Math. Soc., 2000. MR**1774066 (2001h:11001)****3.**T. W. Cusick and M. E. Flahive,*The Markoff and Lagrange spectra*, Math. Surveys and Monograms, no. 30, Amer. Math. Soc., 1989. MR**1010419 (90i:11069)****4.**K. Dajani and C. Kraaikamp,*Ergodic theory of numbers*, The Carus Math. Monograms, no. 29, Math. Assoc. Amer., 2002. MR**1917322 (2003f:37014)****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,*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)**

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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

Email:
abourla@smcm.edu, bourla@american.edu

DOI:
https://doi.org/10.1090/S0002-9939-2013-11601-1

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.