## On the finiteness and periodicity of the $p$-adic Jacobi–Perron algorithm

HTML articles powered by AMS MathViewer

- by
Nadir Murru and Lea Terracini
**HTML**| PDF - Math. Comp.
**89**(2020), 2913-2930 Request permission

## Abstract:

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field of $p$-adic numbers $\mathbb {Q}_p$. MCFs have also been recently introduced in $\mathbb {Q}_p$, including, in particular, a $p$-adic Jacobi–Perron algorithm. In this paper, we address two of the main features of this algorithm, namely its finiteness and periodicity. Regarding the finiteness of the $p$-adic Jacobi–Perron algorithm, our results are obtained by exploiting properties of some auxiliary integer sequences. It is known that a finite $p$-adic MCF represents $\mathbb Q$-linearly dependent numbers. However, we see that the converse is not always true and we prove that in this case infinitely many partial quotients of the MCF have $p$-adic valuations equal to $-1$. Finally, we show that a periodic MCF of dimension $m$ converges to an algebraic irrational of degree less than or equal to $m+1$; for the case $m=2$, we are able to give some more detailed results.## References

- Sami Assaf, Li-Chung Chen, Tegan Cheslack-Postava, Benjamin Cooper, Alexander Diesl, Thomas Garrity, Mathew Lepinski, and Adam Schuyler,
*A dual approach to triangle sequences: a multidimensional continued fraction algorithm*, Integers**5**(2005), no. 1, A8, 39. MR**2192226** - Olga R. Beaver and Thomas Garrity,
*A two-dimensional Minkowski $?(x)$ function*, J. Number Theory**107**(2004), no. 1, 105–134. MR**2059953**, DOI 10.1016/j.jnt.2004.01.008 - 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,
*New infinite classes of periodic Jacobi-Perron algorithms*, Pacific J. Math.**16**(1966), 439–469. MR**190091**, DOI 10.2140/pjm.1966.16.439 - Leon Bernstein,
*The Jacobi-Perron algorithm—Its theory and application*, Lecture Notes in Mathematics, Vol. 207, Springer-Verlag, Berlin-New York, 1971. MR**0285478**, DOI 10.1007/BFb0069405 - 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 - J. B. Coleman,
*A Test for the Type of Irrationality Represented by a Periodic Ternary Continued Fraction*, Amer. J. Math.**52**(1930), no. 4, 835–842. MR**1506786**, DOI 10.2307/2370717 - 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** - 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 - 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 - S. Gerschgorin,
*Über die Abgrenzung der Eigenwerte einer Matrix*, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk, 7, (1931), 749–754. - Fernando Q. Gouvêa,
*$p$-adic numbers*, 2nd ed., Universitext, Springer-Verlag, Berlin, 1997. An introduction. MR**1488696**, DOI 10.1007/978-3-642-59058-0 - 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 - 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**, DOI 10.1017/S1446788700026070 - 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,
*On the periodic writing of cubic irrationals and a generalization of Rédei functions*, Int. J. Number Theory**11**(2015), no. 3, 779–799. MR**3327843**, DOI 10.1142/S1793042115500438 - 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 - Nadir Murru and Lea Terracini,
*On $p$-adic multidimensional continued fractions*, Math. Comp.**88**(2019), no. 320, 2913–2934. MR**3985480**, DOI 10.1090/mcom/3450 - 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** - A. A. Ruban,
*Certain metric properties of the $p$-adic numbers*, Sibirsk. Mat. Ž.**11**(1970), 222–227 (Russian). MR**0260700** - Th. Schneider,
*Über $p$-adische Kettenbrüche*, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 181–189 (German). MR**0272720** - Fritz Schweiger,
*Multidimensional continued fractions*, Oxford Science Publications, Oxford University Press, Oxford, 2000. MR**2121855** - Fritz Schweiger,
*Brun meets Selmer*, Integers**13**(2013), Paper No. A17, 12. MR**3083479** - 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 - Francis Tilborghs,
*Periodic $p$-adic continued fractions*, Simon Stevin**64**(1990), no. 3-4, 383–390. MR**1117188** - B. M. M. de Weger,
*Periodicity of $p$-adic continued fractions*, Elem. Math.**43**(1988), no. 4, 112–116. MR**952010**

## Additional Information

**Nadir Murru**- Affiliation: Department of Mathematics G. Peano, University of Torino, Via Carlo Alberto 10, 10123, Torino, Italy
- MR Author ID: 905269
- Email: nadir.murru@unito.it
**Lea Terracini**- Affiliation: Department of Mathematics G. Peano, University of Torino, Via Carlo Alberto 10, 10123, Torino, Italy
- MR Author ID: 261537
- Email: lea.terracini@unito.it
- Received by editor(s): July 20, 2019
- Received by editor(s) in revised form: January 21, 2020, and February 16, 2020
- Published electronically: May 19, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**89**(2020), 2913-2930 - MSC (2010): Primary 11J70, 12J25, 11J61
- DOI: https://doi.org/10.1090/mcom/3540
- MathSciNet review: 4136551