Necessary and sufficient conditions for convergence of integer continued fractions
HTML articles powered by AMS MathViewer
- by Ian Short and Margaret Stanier PDF
- Proc. Amer. Math. Soc. 150 (2022), 617-631 Request permission
Abstract:
Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational.
An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is a graph embedded in the hyperbolic plane that induces a tessellation of the hyperbolic plane by ideal triangles. With this geometric representation of continued fractions our test for convergence can be interpreted in a particularly elegant manner, giving deeper insight into the nature of continued fraction convergence.
References
- A. F. Beardon, M. Hockman, and I. Short, Geodesic continued fractions, Michigan Math. J. 61 (2012), no. 1, 133–150. MR 2904005, DOI 10.1307/mmj/1331222851
- Douglas Bowman and J. Mc Laughlin, Continued fractions with multiple limits, Adv. Math. 210 (2007), no. 2, 578–606. MR 2303233, DOI 10.1016/j.aim.2006.07.004
- S. G. Dani, Continued fraction expansions for complex numbers—a general approach, Acta Arith. 171 (2015), no. 4, 355–369. MR 3430769, DOI 10.4064/aa171-4-4
- S. G. Dani and Arnaldo Nogueira, Continued fractions for complex numbers and values of binary quadratic forms, Trans. Amer. Math. Soc. 366 (2014), no. 7, 3553–3583. MR 3192607, DOI 10.1090/S0002-9947-2014-06003-0
- Marius Iosifescu and Cor Kraaikamp, Metrical theory of continued fractions, Mathematics and its Applications, vol. 547, Kluwer Academic Publishers, Dordrecht, 2002. MR 1960327, DOI 10.1007/978-94-015-9940-5
- Oleg Karpenkov, Geometry of continued fractions, Algorithms and Computation in Mathematics, vol. 26, Springer, Heidelberg, 2013. MR 3099298, DOI 10.1007/978-3-642-39368-6
- Svetlana Katok, Coding of closed geodesics after Gauss and Morse, Geom. Dedicata 63 (1996), no. 2, 123–145. MR 1413625, DOI 10.1007/BF00148213
- Svetlana Katok and Ilie Ugarcovici, Geometrically Markov geodesics on the modular surface, Mosc. Math. J. 5 (2005), no. 1, 135–155 (English, with English and Russian summaries). MR 2153471, DOI 10.17323/1609-4514-2005-5-1-135-155
- Svetlana Katok and Ilie Ugarcovici, Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 87–132. MR 2265011, DOI 10.1090/S0273-0979-06-01115-3
- Lisa Lorentzen, Convergence of random continued fractions and random iterations of Möbius transformations, Modern trends in constructive function theory, Contemp. Math., vol. 661, Amer. Math. Soc., Providence, RI, 2016, pp. 57–71. MR 3489550, DOI 10.1090/conm/661/13274
- Lisa Lorentzen and Haakon Waadeland, Continued fractions. Vol. 1, 2nd ed., Atlantis Studies in Mathematics for Engineering and Science, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. Convergence theory. MR 2433845, DOI 10.2991/978-94-91216-37-4
- Oskar Perron, Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954 (German). 3te Aufl. MR 0064172
- Oskar Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957 (German). MR 0085349
- Ian Short and Mairi Walker, Geodesic Rosen continued fractions, Q. J. Math. 67 (2016), no. 4, 519–549. MR 3609844, DOI 10.1093/qmath/haw025
Additional Information
- Ian Short
- Affiliation: School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, United Kingdom
- MR Author ID: 791601
- ORCID: 0000-0002-7360-4089
- Margaret Stanier
- Affiliation: School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, United Kingdom
- ORCID: 0000-0002-6010-2864
- Received by editor(s): October 16, 2019
- Received by editor(s) in revised form: February 19, 2021
- Published electronically: November 4, 2021
- Communicated by: Matthew A. Papanikolas
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 617-631
- MSC (2020): Primary 40A15; Secondary 11A55
- DOI: https://doi.org/10.1090/proc/15574
- MathSciNet review: 4356172