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An unusual continued fraction


Authors: Dzmitry Badziahin and Jeffrey Shallit
Journal: Proc. Amer. Math. Soc. 144 (2016), 1887-1896
MSC (2010): Primary 11J70, 11J82; Secondary 11Y65, 11A55
DOI: https://doi.org/10.1090/proc/12848
Published electronically: September 15, 2015
MathSciNet review: 3460151
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Abstract: We consider the real number $ \sigma $ with continued fraction expansion $ [a_0, a_1, a_2,\ldots ] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots ]$, where $ a_i$ is the largest power of $ 2$ dividing $ i+1$. We show that the irrationality measure of $ \sigma ^2$ is at least $ 8/3$. We also show that certain partial quotients of $ \sigma ^2$ grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.


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  • [1] Boris Adamczewski and Jean-Paul Allouche, Reversals and palindromes in continued fractions, Theoret. Comput. Sci. 380 (2007), no. 3, 220-237. MR 2330994 (2008d:11074), https://doi.org/10.1016/j.tcs.2007.03.017
  • [2] Boris Adamczewski and Yann Bugeaud, On the complexity of algebraic numbers. II. Continued fractions, Acta Math. 195 (2005), 1-20. MR 2233683 (2008b:11080), https://doi.org/10.1007/BF02588048
  • [3] Boris Adamczewski and Yann Bugeaud, A short proof of the transcendence of Thue-Morse continued fractions, Amer. Math. Monthly 114 (2007), no. 6, 536-540. MR 2321257 (2008d:11072)
  • [4] Boris Adamczewski and Yann Bugeaud, Palindromic continued fractions, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1557-1574 (English, with English and French summaries). MR 2364142 (2008m:40009)
  • [5] Boris Adamczewski and Yann Bugeaud, Transcendence measures for continued fractions involving repetitive or symmetric patterns, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 883-914. MR 2654083 (2011g:11138), https://doi.org/10.4171/JEMS/218
  • [6] J.-P. Allouche, J. L. Davison, M. Queffélec, and L. Q. Zamboni, Transcendence of Sturmian or morphic continued fractions, J. Number Theory 91 (2001), no. 1, 39-66. MR 1869317 (2002k:11117), https://doi.org/10.1006/jnth.2001.2669
  • [7] Jean-Paul Allouche and Jeffrey Shallit, The ring of $ k$-regular sequences, Theoret. Comput. Sci. 98 (1992), no. 2, 163-197. MR 1166363 (94c:11021), https://doi.org/10.1016/0304-3975(92)90001-V
  • [8] Jean-Paul Allouche and Jeffrey Shallit, The ring of $ k$-regular sequences. II, Theoret. Comput. Sci. 307 (2003), no. 1, 3-29. Words. MR 2014728 (2004m:68172), https://doi.org/10.1016/S0304-3975(03)00090-2
  • [9] J. Borwein, A. van der Poorten, J. Shallit, and W. Zudilin,
    Neverending Fractions: An Introduction to Continued Fractions.
    Cambridge University Press, 2014.
  • [10] Yann Bugeaud, Continued fractions with low complexity: Transcendence measures and quadratic approximation, Compos. Math. 148 (2012), no. 3, 718-750. MR 2925396, https://doi.org/10.1112/S0010437X11007524
  • [11] Yann Bugeaud, Automatic continued fractions are transcendental or quadratic, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 6, 1005-1022 (English, with English and French summaries). MR 3134686
  • [12] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708 (19,396h)
  • [13] J. S. Frame, Classroom Notes: Continued Fractions and Matrices, Amer. Math. Monthly 56 (1949), no. 2, 98-103. MR 1527170, https://doi.org/10.2307/2306169
  • [14] G. H. Hardy and E. M. Wright,
    An Introduction to the Theory of Numbers,
    Oxford University Press, 5th edition, 1985.
  • [15] A. Hurwitz, Über die Kettenbrüche, deren Teilnenner arithmetische Reihen bilden. Vierteljahrsschrift d. Naturforsch Gesellschaft in Zürich, Jahrg. 41 (1896),
    In Mathematische Werke, Band II, Birkhäuser, Basel, 1963, pp. 276-302.
  • [16] Adolf Hurwitz, Lectures on number theory, Universitext, Springer-Verlag, New York, 1986. Translated from the German and with a preface by William C. Schulz; Translation edited and with a preface by Nikolaos Kritikos. MR 816942 (88f:11002)
  • [17] Kjell Kolden, Continued fractions and linear substitutions, Arch. Math. Naturvid. 50 (1949), no. 6, 141-196. MR 0032033 (11,244a)
  • [18] E. Maillet,
    Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions.
    Gauthier-Villars, 1906.
  • [19] K. R. Matthews and R. F. C. Walters, Some properties of the continued fraction expansion of $ (m/n)e^{1/q}$, Proc. Cambridge Philos. Soc. 67 (1970), 67-74. MR 0252889 (40 #6104)
  • [20] Oskar Perron, Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954 (German). 3te Aufl. MR 0064172 (16,239e)
  • [21] M. Queffélec, Transcendance des fractions continues de Thue-Morse, J. Number Theory 73 (1998), no. 2, 201-211 (French, with French summary). MR 1658023 (99j:11081), https://doi.org/10.1006/jnth.1998.2305
  • [22] Martine Queffélec, Irrational numbers with automaton-generated continued fraction expansion, Dynamical systems (Luminy-Marseille, 1998) World Sci. Publ., River Edge, NJ, 2000, pp. 190-198. MR 1796159 (2001h:11008)
  • [23] K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20; corrigendum, 168. MR 0072182 (17,242d)
  • [24] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,
    Available at http://oeis.org.
  • [25] R. F. C. Walters, Alternative derivation of some regular continued fractions, J. Austral. Math. Soc. 8 (1968), 205-212. MR 0226245 (37 #1835)

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Additional Information

Dzmitry Badziahin
Affiliation: Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham, DH1 3LE, United Kingdom
Email: dzmitry.badziahin@durham.ac.uk

Jeffrey Shallit
Affiliation: School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: shallit@cs.uwaterloo.ca

DOI: https://doi.org/10.1090/proc/12848
Received by editor(s): May 4, 2015
Received by editor(s) in revised form: May 26, 2015
Published electronically: September 15, 2015
Additional Notes: The research of the first author was supported by EPSRC Grant EP/L005204/1.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

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