An effective criterion for periodicity of $\ell$-adic continued fractions
HTML articles powered by AMS MathViewer
- by Laura Capuano, Francesco Veneziano and Umberto Zannier;
- Math. Comp. 88 (2019), 1851-1882
- DOI: https://doi.org/10.1090/mcom/3385
- Published electronically: October 18, 2018
- HTML | PDF | Request permission
Abstract:
The theory of continued fractions has been generalized to $\ell$-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic irrationals for the $\ell$-adic continued fractions introduced by Ruban. In this case, rational numbers may have a periodic non-terminating continued fraction expansion; moreover, for quadratic irrational numbers, no analogue of Lagrange’s theorem holds. We give general explicit criteria to establish the periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol.References
- Edmondo Bedocchi, Fractions continues $p$-adiques: périodes de longueur paire, Boll. Un. Mat. Ital. A (7) 7 (1993), no. 2, 259–265 (French, with Italian summary). MR 1234077
- Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
- Jerzy Browkin, Continued fractions in local fields. I, Demonstratio Math. 11 (1978), no. 1, 67–82. MR 506059
- Jerzy Browkin, Continued fractions in local fields. II, Math. Comp. 70 (2001), no. 235, 1281–1292. MR 1826582, DOI 10.1090/S0025-5718-00-01296-5
- P. Bundschuh, $p$-adische Kettenbrüche und Irrationalität $p$-adischer Zahlen, Elem. Math. 32 (1977), no. 2, 36–40 (German). MR 453620
- Doug Hensley, Continued fractions, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2351741, DOI 10.1142/9789812774682
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
- Vichian Laohakosol, A characterization of rational numbers by $p$-adic Ruban continued fractions, J. Austral. Math. Soc. Ser. A 39 (1985), no. 3, 300–305. MR 802720
- K. Mahler, Zur approximation p-adischer irrationalzahlen, Nieuw Arch. Wisk. 2 (1934), no. 18, 22–34.
- Tomohiro Ooto, Transcendental $p$-adic continued fractions, Math. Z. 287 (2017), no. 3-4, 1053–1064. MR 3719527, DOI 10.1007/s00209-017-1859-2
- A. A. Ruban, Certain metric properties of the $p$-adic numbers, Sibirsk. Mat. Ž. 11 (1970), 222–227 (Russian). MR 260700
- Th. Schneider, Über $p$-adische Kettenbrüche, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London-New York, 1970, pp. 181–189 (German). MR 272720
- Lian Xiang Wang, $p$-adic continued fractions. I, II, Sci. Sinica Ser. A 28 (1985), no. 10, 1009–1017, 1018–1023. MR 866457
Bibliographic Information
- Laura Capuano
- Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
- MR Author ID: 1146921
- Email: Laura.Capuano@maths.ox.ac.uk
- Francesco Veneziano
- Affiliation: Centro di Ricerca Matematica Ennio De Giorgi, Piazza dei Cavalieri, 3, 56126 Pisa, Italy
- MR Author ID: 966417
- ORCID: 0000-0002-2225-7769
- Email: francesco.veneziano@sns.it
- Umberto Zannier
- Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
- MR Author ID: 186540
- Email: umberto.zannier@sns.it
- Received by editor(s): January 29, 2018
- Received by editor(s) in revised form: May 30, 2018
- Published electronically: October 18, 2018
- Additional Notes: The first author was funded by the INdAM [Borsa Ing. G. Schirillo], the European Research Council [267273] and the Engineering and Physical Sciences Research Council [EP/N007956/1].
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1851-1882
- MSC (2010): Primary 11J70, 11D88, 11Y16
- DOI: https://doi.org/10.1090/mcom/3385
- MathSciNet review: 3925488