Tokyo Journal of Mathematics

On the Construction of Continued Fraction Normal Series in Positive Characteristic

Dong Han KIM, Hitoshi NAKADA, and Rie NATSUI

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Motivated by the famous Champernowne construction of a normal number, R.~Adler, M.~Keane, and M.~Smorodinsky constructed a normal number with respect to the simple continued fraction transformation. In this paper, we follow their idea and construct a normal series for the Artin continued fraction expansion in positive characteristic. A normal series for L\"uroth expansion is also discussed.

Article information

Tokyo J. Math., Volume 39, Number 3 (2017), 679-694.

First available in Project Euclid: 6 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]
Secondary: 11J70: Continued fractions and generalizations [See also 11A55, 11K50]


KIM, Dong Han; NAKADA, Hitoshi; NATSUI, Rie. On the Construction of Continued Fraction Normal Series in Positive Characteristic. Tokyo J. Math. 39 (2017), no. 3, 679--694. doi:10.3836/tjm/1475723093.

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  • R. Adler, M. Keane and M. Smorodinsky, A construction of a normal number for the continued fraction transformation, J. Number Theory 13 (1981), 95–105.
  • E. Artin, Ein mechanisches system mit quasiergodischen bahnen, Abh. Math. Sem. Univ. Hamburg 3 (1924), 170–175.
  • V. Berthé and H. Nakada, On continued fraction expansions in positive characteristic: equivalence relations and some metric properties, Expo. Math. 18 (2000), 257–284.
  • D. G. Champernowne, The construction of decimal normal in the scale of ten, J. London Math. Soc. 8 (1933), 254–260.
  • K. Inoue and H. Nakada, On metric Diophantine approximation in positive characteristic, Acta Arith. 110 (2003), 205–218.
  • J. Knopfmacher, Ergodic properties of some inverse polynomial series expansions of Laurent series, Acta Math. Hungar. 60 (1992), 241–246.
  • A. Knopfmacher and J. Knopfmacher, Inverse polynomial expansions of Laurent series, Constr. Approx. 4 (1988), 379–389.
  • A. Knopfmacher and J. Knopfmacher, Metric properties of algorithms inducing Lüroth series expansions of Laurent series, Astérisque 15 (1992), 237–246.
  • S. Kristensen, Some metric properties of Lüroth expansions over the field of Laurent series, Bull. Austral. Math. Soc. 64 (2001), 345–351.
  • R. Paysant-Leroux and E. Dubois, Ètude mètrique de l'algorithme de Jacobi-Perron dans un corps de sèries formelles, C. R. Acad. Sci. Paris Ser. A-B 275 (1972), A683–A686.
  • W. M. Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arith. 95 (2000), 139–166.