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An effective criterion for periodicity of $ \ell$-adic continued fractions


Authors: Laura Capuano, Francesco Veneziano and Umberto Zannier
Journal: Math. Comp. 88 (2019), 1851-1882
MSC (2010): Primary 11J70, 11D88, 11Y16
DOI: https://doi.org/10.1090/mcom/3385
Published electronically: October 18, 2018
MathSciNet review: 3925488
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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.


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Additional Information

Laura Capuano
Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
Email: Laura.Capuano@maths.ox.ac.uk

Francesco Veneziano
Affiliation: Centro di Ricerca Matematica Ennio De Giorgi, Piazza dei Cavalieri, 3, 56126 Pisa, Italy
Email: francesco.veneziano@sns.it

Umberto Zannier
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email: umberto.zannier@sns.it

DOI: https://doi.org/10.1090/mcom/3385
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].
Article copyright: © Copyright 2018 American Mathematical Society