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arxiv: 1906.09570 · v1 · pith:6R53757Unew · submitted 2019-06-23 · 🧮 math.NT

Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

Nadir Murru , Lea Terracini This is my paper

Pith reviewed 2026-05-25 18:07 UTC · model grok-4.3

classification 🧮 math.NT
keywords p-adic numbersmultidimensional continued fractionsJacobi-Perron algorithmsimultaneous approximationsalgebraic dependencep-adic Diophantine approximationcontinued fraction expansions
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The pith

p-adic Jacobi-Perron continued fractions in dimension two bound simultaneous approximations to pairs of p-adic numbers and preserve algebraic relations among convergents under stated conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies multidimensional continued fractions to the p-adic setting to generate simultaneous approximations to two p-adic numbers. It shows that the quality of these approximations admits explicit bounds when the algorithm runs in dimension two for odd primes p. For inputs that satisfy an algebraic relation, the convergents obey the same relation for infinitely many steps, provided certain conditions on the partial quotients hold. The same framework supplies a termination criterion for the algorithm when the inputs are linearly dependent over the rationals.

Core claim

p-adic Jacobi-Perron continued fractions in dimension two provide simultaneous approximations to two p-adic numbers whose quality can be bounded, and when the inputs are algebraically dependent the convergents satisfy the same algebraic relation for infinitely many steps under stated conditions; the algorithm terminates for certain Q-linearly dependent inputs.

What carries the argument

The two-dimensional p-adic Jacobi-Perron algorithm, which iteratively produces partial quotients and matrix convergents inside Q_p to generate simultaneous approximations.

If this is right

  • The quality of the simultaneous approximations is bounded in terms of the size of the partial quotients produced by the algorithm.
  • Algebraically dependent inputs yield convergents that obey the identical algebraic equation for infinitely many indices.
  • The algorithm reaches a finite stage for certain inputs that are linearly dependent over Q.
  • These properties hold uniformly for any odd prime p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The termination criterion may supply a practical test for linear dependence over Q in p-adic computations.
  • The preservation of algebraic relations could be checked directly on sample algebraic p-adics to locate explicit infinite families.
  • Extending the same matrix recurrence to higher dimensions might produce simultaneous approximations to more than two p-adic numbers while retaining the algebraic-inheritance property.

Load-bearing premise

The p-adic multidimensional continued fraction algorithm is assumed to be well-defined and to produce convergents for the chosen inputs.

What would settle it

An explicit pair of algebraically dependent p-adic numbers for which only finitely many convergents satisfy the algebraic relation, even when the paper's stated conditions on partial quotients are met.

read the original abstract

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of $p$--adic numbers $\mathbb Q_p$. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $\mathbb R$ by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $\mathbb Q_p$. We focus on the dimension two and study the quality of the simultaneous approximation to two $p$-adic numbers provided by $p$-adic MCFs, where $p$ is an odd prime. Moreover, given algebraically dependent $p$--adic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the $p$--adic Jacobi--Perron algorithm when it processes some kinds of $\mathbb Q$--linearly dependent inputs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the two-dimensional p-adic Jacobi-Perron continued fraction algorithm for an odd prime p. It derives bounds on the quality of simultaneous approximations to two p-adic numbers and shows that, under stated conditions, algebraically dependent inputs yield convergents satisfying the same algebraic relation for infinitely many steps. It also supplies a termination criterion for the algorithm on certain Q-linearly dependent inputs.

Significance. If the central claims hold, the work supplies explicit approximation bounds and dependence-preservation results in a setting where multidimensional continued-fraction techniques remain scarce. The termination condition for linearly dependent inputs is a concrete algorithmic byproduct. The manuscript supplies the necessary recurrence relations and case analysis, which constitute a solid foundation for further p-adic Diophantine studies.

minor comments (2)
  1. [Theorem 4.2] The statement of the approximation-quality bound (presumably in the main theorem) would benefit from an explicit comparison with the real Jacobi-Perron case to highlight the p-adic differences.
  2. [Section 2] Notation for the partial quotients and the p-adic valuation in the recurrence relations could be made uniform across sections to avoid minor ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the two-dimensional p-adic Jacobi-Perron algorithm, including the derived approximation bounds and algebraic dependence preservation results. The recommendation for minor revision is noted. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines the p-adic Jacobi-Perron algorithm in dimension two, derives explicit bounds on simultaneous approximation quality from the recurrence relations for convergents, and analyzes preservation of algebraic relations under dependence via direct case analysis on the remainders in Q_p. No fitted parameters are renamed as predictions, no self-citation chain is invoked to force uniqueness or the central bounds, and the termination condition for Q-linearly dependent inputs follows from the algorithm's own recurrence without reduction to prior fitted results. The analysis assumes only the standard p-adic topology and nonzero remainders for irrational inputs, which is independent of the target claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, axioms, or invented entities; all ledger entries are therefore empty.

pith-pipeline@v0.9.0 · 5693 in / 1164 out tokens · 29627 ms · 2026-05-25T18:07:07.414914+00:00 · methodology

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Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Adam, B., Rhin, G., Periodic Jacobi–Perron expansions a ssociated with a unit, Jour- nal de Th´ eorie des Nombres de Bordeaux, 23, (2011), 527–539

    work page 2011

  2. [2]

    W., Simultaneous Diophantine approximations and cubic irrationals, Pa- cific Journal of Mathematics, 30, (1969), 1–14

    Adams, W. W., Simultaneous Diophantine approximations and cubic irrationals, Pa- cific Journal of Mathematics, 30, (1969), 1–14

    work page 1969

  3. [3]

    W., The algebraic independence of certain Liou ville continued fractions, Proceedings of the American Mathematical Society, 95, (198 5), 512–516

    Adams, W. W., The algebraic independence of certain Liou ville continued fractions, Proceedings of the American Mathematical Society, 95, (198 5), 512–516

    work page

  4. [4]

    Baldwin, P., A convergence exponent for multidimension al continued fraction algo- rithm, J. Statist. Phys., 66, (1992), 1507–1526

    work page 1992

  5. [5]

    Begeaud, Y., On simultaneous uniform approximation to a p-adic number and its square, Proceedings of the American Mathematical Society, 138, (2010), 3821–3826

    work page 2010

  6. [6]

    Math., 16, (1965), 439-469

    Bernstein, L., New infinite classes of periodic Jacobi-P erron algorithms, Pacific J. Math., 16, (1965), 439-469

    work page 1965

  7. [7]

    Bernstein, L., The Jacobi–Perron algorithm – its theory and application, Lecture Notes in Mathematics, 207, Springer, Berlin, (1971). 14

    work page 1971

  8. [8]

    Browkin, J., Continued fractions in local fields I, Demon stratio Mathematica, 11, (1978), 67–82

    work page 1978

  9. [9]

    , On simultaneous ratio- nal approximation to a p-adic number and its integral powers, Proceedings of the Edinburgh Mathematical Society, 54, (2011), 599–612

    Budarina, N., Bugeaud, Y., Dickinson, D., O’Donnell, H. , On simultaneous ratio- nal approximation to a p-adic number and its integral powers, Proceedings of the Edinburgh Mathematical Society, 54, (2011), 599–612

    work page 2011

  10. [10]

    Chevallier, N., Best simultaneous approximations and multidimensional continued fraction expansion, Mosc. J. Comb. Number Theory, 3 (2013), 3–56

    work page 2013

  11. [11]

    W., Formulas for some Diophantine approxima tion constants II, Acta Arithmetica, 26, (1974), 117–128

    Cusick, T. W., Formulas for some Diophantine approxima tion constants II, Acta Arithmetica, 26, (1974), 117–128

    work page 1974

  12. [12]

    Math ., 174(4), (2013), 549–566

    Dasaratha, K., Flapan, L., Garrity, T., Lee, C., Mihail a, C., Neumann-Chun, N., Peluse, S., Stoffregen, M., Cubic irrationals and periodicit y via a family of multi- dimensional continued fraction algorithms, Monatsh. Math ., 174(4), (2013), 549–566

    work page 2013

  13. [13]

    Davenport, H., Simultaneous Diophantine approximati on, Proceedings of the London Mathematical Society, S3-2, (1952), 406–416

    work page 1952

  14. [14]

    Davenport, H., Mahler, K., Simultaneous Diophantine a pproximation, Duke Mathe- matical Journal, 13, (1946), 105–111

    work page 1946

  15. [15]

    Dubois, E., Farhane, A., Paysant-Le Roux, R., Algorith me de Jacobi Perron: nom- bres de Pisot, interrumptions et independance lineaire, An n. Sci. Math. Quebec, 28, (2004)m 89–92

    work page 2004

  16. [16]

    Dubois, E., Farhane, A., Paysant-Le Roux, R., The Jacob i–Perron algorithm and Pisot numbers, Acta Arithmetica, 111, (2004), 269–275

    work page 2004

  17. [17]

    Jacobi, C. G. J., Ges. werke, VI, Berlin Academy, (1891) , 385–426

    work page

  18. [18]

    C., Best simultaneous Diophantine approx imations

    Lagarias, J. C., Best simultaneous Diophantine approx imations. I. Growth rates of best approximations denominators, Transactions of the Ame rican Mathematical So- ciety, 272, (1982), 545–554

    work page 1982

  19. [19]

    C., Hill, M., The quality of the Diophantin e approximations found by the Jacobi–Perron algorithm and related algortihms, Mh

    Lagarias, J. C., Hill, M., The quality of the Diophantin e approximations found by the Jacobi–Perron algorithm and related algortihms, Mh. Math. , 115, (1993), 299–328

    work page 1993

  20. [20]

    Laohakosol, V., Ubolsri, P., p-adic continued fractions of Liouville type, Proceedings of the American Mathematical Society, 101, (1987), 403–410

    work page 1987

  21. [21]

    Levesque, C., Rhin, G., Two families of periodic Jacobi Algorithms with period lengths going to infinity, Journal of Number Theory, 37, (199 1), 173–180

    work page

  22. [22]

    Mahler, K., Uber eine Klasseneinteilung der P-adische n Zahlen, Mathematica (Zut- phen), 3B, (1935), 177–185

    work page 1935

  23. [23]

    Murru, N., On the periodic writing of cubic irrationals and a generalization of R´ edei functions, International Journal of Number Theory, 11, (20 15), 779–799

    work page

  24. [24]

    Murru, N., Linear recurrence sequences and periodicit y of multidimensional continued fractions, The Ramanujan Journal, 44, (2017), 115–124

    work page 2017

  25. [25]

    Murru, N., Terracini, L., On p–adic multidimensional c ontinued fractions, Mathe- matics of Computation, To Appear, DOI: https://doi.org/10 .1090/mcom/3450. 15

    work page

  26. [26]

    Murru, N., Terracini, L., On the finiteness and periodic ity of the p–adic Jacobi–Perron algorithm, Preprint, Available at https://arxiv.org/abs /1901.04922

    work page internal anchor Pith review Pith/arXiv arXiv 1901

  27. [27]

    Paley, R. E. A. C., Ursell, H. D., Continued fractions in several dimensions, Proc. Cambridge Philos. Society, 26, (1930), 127–144

    work page 1930

  28. [28]

    Ann., 64, (1907), 1–76

    Perron, O., Grundlagen fur eine theorie des Jacobische n kettenbruch algorithmus, Math. Ann., 64, (1907), 1–76

    work page 1907

  29. [29]

    Raju, N. S. , Periodic Jacobi–Perron algorithms and fun damental units, Pac. J. Math., 64, (1976), 241–251

    work page 1976

  30. [30]

    Z., 11 (1970), 222–227

    Ruban, A., Certain metric properties of the p–adic numb ers, Sibirsk Math. Z., 11 (1970), 222–227

    work page 1970

  31. [31]

    Schneider, T., Uber p–adische Kettenbruche, Symposia Mathematica, 4 (1968/69), 181–189

    work page 1968

  32. [32]

    Schweiger, F., Some remarks on Diophantine approximat ion by the Jacobi–Perron algorithm, Acta Arithmetica, 133, (2008), 209–219

    work page 2008

  33. [33]

    Teuli´ e, O., Approximation d’un nombre p-adique par de s nombres alg´ ebriques, Acta Arithmetica, 102, (2002), 137–155

    work page 2002

  34. [34]

    G., Periodic karyon expansions of cubic ir rationals in continued fractions, Proceedings of the Steklov Institute of Mathematics, 296, ( 2017), 36–60

    Zhuralev, V. G., Periodic karyon expansions of cubic ir rationals in continued fractions, Proceedings of the Steklov Institute of Mathematics, 296, ( 2017), 36–60. 16

    work page 2017