On $p$-adic multidimensional continued fractions
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- by Nadir Murru and Lea Terracini;
- Math. Comp. 88 (2019), 2913-2934
- DOI: https://doi.org/10.1090/mcom/3450
- Published electronically: May 17, 2019
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Abstract:
Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the $p$-adic numbers $\mathbb Q_p$. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of an MCF, and we perform a general study about their convergence in $\mathbb Q_p$. In particular, we derive some sufficient conditions for their convergence and we prove that convergent MCFs always strongly converge in $\mathbb Q_p$ contrary to the real case where strong convergence is not always guaranteed. Then, we focus on a specific algorithm that, starting from an $m$-tuple of numbers in $\mathbb Q_p$ ($p$ odd), produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized $p$-adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.References
- Edmondo Bedocchi, A note on $p$-adic continued fractions, Ann. Mat. Pura Appl. (4) 152 (1988), 197–207 (Italian, with English summary). MR 980980, DOI 10.1007/BF01766149
- Edmondo Bedocchi, Sur le développement de $\sqrt m$ en fraction continue $p$-adique, Manuscripta Math. 67 (1990), no. 2, 187–195 (French, with English summary). MR 1042237, DOI 10.1007/BF02568429
- Leon Bernstein, The Jacobi-Perron algorithm—Its theory and application, Lecture Notes in Mathematics, Vol. 207, Springer-Verlag, Berlin-New York, 1971. MR 285478
- A. D. Bryuno and V. I. Parusnikov, Comparison of various generalizations of continued fractions, Mat. Zametki 61 (1997), no. 3, 339–348 (Russian, with Russian summary); English transl., Math. Notes 61 (1997), no. 3-4, 278–286. MR 1619743, DOI 10.1007/BF02355409
- 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
- Laura Capuano, Francesco Veneziano, and Umberto Zannier, An effective criterion for periodicity of $\ell$-adic continued fractions, Math. Comp. 88 (2019), no. 318, 1851–1882. MR 3925488, DOI 10.1090/mcom/3385
- Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, and Matthew Stoffregen, A generalized family of multidimensional continued fractions: TRIP Maps, Int. J. Number Theory 10 (2014), no. 8, 2151–2186. MR 3273480, DOI 10.1142/S1793042114500730
- Eugène Dubois, Ahmed Farhane, and Roger Paysant-Le Roux, Étude des interruptions dans l’algorithme de Jacobi-Perron, Bull. Austral. Math. Soc. 69 (2004), no. 2, 241–254 (French, with English summary). MR 2051360, DOI 10.1017/S000497270003598X
- Eugène Dubois and Roger Paysant Le Roux, Développement périodique par l’algorithme de Jacobi-Perron et nombre de Pisot-Vijayaraghvan, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A649–A652 (French). MR 285480
- Eric Errthum, A division algorithm approach to $p$-adic Sylvester expansions, J. Number Theory 160 (2016), 1–10. MR 3425194, DOI 10.1016/j.jnt.2015.08.016
- Thomas Garrity, On periodic sequences for algebraic numbers, J. Number Theory 88 (2001), no. 1, 86–103. MR 1825992, DOI 10.1006/jnth.2000.2608
- O. N. German and E. L. Lakshtanov, On a multidimensional generalization of Lagrange’s theorem for continued fractions, Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), no. 1, 51–66 (Russian, with Russian summary); English transl., Izv. Math. 72 (2008), no. 1, 47–61. MR 2394971, DOI 10.1070/IM2008v072n01ABEH002391
- J. Hančl, A. Jaššová, P. Lertchoosakul, and R. Nair, On the metric theory of $p$-adic continued fractions, Indag. Math. (N.S.) 24 (2013), no. 1, 42–56. MR 2997750, DOI 10.1016/j.indag.2012.06.004
- M. D. Hendy and N. S. Jeans, The Jacobi-Perron algorithm in integer form, Math. Comp. 36 (1981), no. 154, 565–574. MR 606514, DOI 10.1090/S0025-5718-1981-0606514-X
- C. Hermite, Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objects de la théorie des nombres, J. Reine Angew. Math. 40 (1850), 261–278 (French). MR 1578698, DOI 10.1515/crll.1850.40.261
- C. G. J. Jacobi, Ges. Werke VI, Berlin Academy, (1891), 385–426.
- O. N. Karpenkov, Constructing multidimensional periodic continued fractions in the sense of Klein, Math. Comp. 78 (2009), no. 267, 1687–1711. MR 2501070, DOI 10.1090/S0025-5718-08-02187-X
- Michitaka Kojima, Continued fractions in $p$-adic numbers, Algebraic number theory and related topics 2010, RIMS Kôkyûroku Bessatsu, B32, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012, pp. 239–254. MR 2986927
- C. Lager, A p-adic Euclidean algorithm, Rose–Hulman Undergraduate Mathematics Journal, 10 (2009), Article 9.
- 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
- Justin Miller, On p-adic continued fractions and quadratic irrationals, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–The University of Arizona. MR 2710960
- Nadir Murru, Linear recurrence sequences and periodicity of multidimensional continued fractions, Ramanujan J. 44 (2017), no. 1, 115–124. MR 3696138, DOI 10.1007/s11139-016-9820-2
- Kentaro Nakaishi, Strong convergence of additive multidimensional continued fraction algorithms, Acta Arith. 121 (2006), no. 1, 1–19. MR 2216301, DOI 10.4064/aa121-1-1
- 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
- Oskar Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus, Math. Ann. 64 (1907), no. 1, 1–76 (German). MR 1511422, DOI 10.1007/BF01449880
- A. J. van der Poorten, Schneider’s continued fraction, Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991) Lecture Notes in Pure and Appl. Math., vol. 147, Dekker, New York, 1993, pp. 271–281. MR 1219341
- Nambury S. Raju, Periodic Jacobi-Perron algorithms and fundamental units, Pacific J. Math. 64 (1976), no. 1, 241–251. MR 424746
- A. A. Ruban, Certain metric properties of the $p$-adic numbers, Sibirsk. Mat. Ž. 11 (1970), 222–227 (Russian). MR 260700
- A. Saito, J. Tamura, S. Yasutomi, Continued fractions algorithm and Lagrange’s theorem in $\mathbb Q_p$, available at https://arxiv.org/abs/1701.04615v1, (2017).
- A. Saito, J. Tamura, S. Yasutomi, Multidimensional p-adic continued fraction algorithms, available at https://arxiv.org/abs/1705.06122, (2017).
- 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
- Fritz Schweiger, Multidimensional continued fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000. MR 2121855
- Jun-ichi Tamura and Shin-ichi Yasutomi, A new multidimensional continued fraction algorithm, Math. Comp. 78 (2009), no. 268, 2209–2222. MR 2521286, DOI 10.1090/S0025-5718-09-02217-0
- Jun-ichi Tamura, A $p$-adic phenomenon related to certain integer matrices, and $p$-adic values of a multidimensional continued fraction, Summer School on the Theory of Uniform Distribution, RIMS Kôkyûroku Bessatsu, B29, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012, pp. 1–40. MR 2962866
- Francis Tilborghs, Periodic $p$-adic continued fractions, Simon Stevin 64 (1990), no. 3-4, 383–390. MR 1117188
- Vichian Laohakosol and Patchara Ubolsri, $p$-adic continued fractions of Liouville type, Proc. Amer. Math. Soc. 101 (1987), no. 3, 403–410. MR 908638, DOI 10.1090/S0002-9939-1987-0908638-3
- Lian Xiang Wang, $p$-adic continued fractions. I, II, Sci. Sinica Ser. A 28 (1985), no. 10, 1009–1017, 1018–1023. MR 866457
- B. M. M. de Weger, Periodicity of $p$-adic continued fractions, Elem. Math. 43 (1988), no. 4, 112–116. MR 952010
Bibliographic Information
- Nadir Murru
- Affiliation: Department of Mathematics L. Lagrange, Politecnico of Turin, Turin, Italy
- MR Author ID: 905269
- Email: nadir.murru@gmail.com
- Lea Terracini
- Affiliation: Department of Mathematics G. Peano, University of Turin, Turin, Italy
- MR Author ID: 261537
- Email: lea.terracini@unito.it
- Received by editor(s): April 20, 2018
- Received by editor(s) in revised form: October 31, 2018, and January 21, 2019
- Published electronically: May 17, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 2913-2934
- MSC (2010): Primary 11J70, 12J25, 11J61
- DOI: https://doi.org/10.1090/mcom/3450
- MathSciNet review: 3985480