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λŒ€ν•œμˆ˜ν•™νšŒ (Korean Mathematical Society)
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ON THE PERIOD OF Ξ²-EXPANSION OF PISOT OR SALEM SERIES OVER 𝔽q((x-1))

  • RIM, GHORBEL (FACULTE DES SCIENCES DE SFAX DEPARTEMENT DE MATHEMATIQUES) ;
  • SOUROUR, ZOUARI (FACULTE DES SCIENCES DE SFAX DEPARTEMENT DE MATHEMATIQUES)
  • 투고 : 2013.03.15
  • λ°œν–‰ : 2015.07.31
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초둝

In [6], it is proved that the lengths of periods occurring in the ${\beta}$-expansion of a rational series r noted by $Per_{\beta}(r)$ depend only on the denominator of the reduced form of r for quadratic Pisot unit series. In this paper, we will show first that every rational r in the unit disk has strictly periodic ${\beta}$-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if $r=\frac{P}{Q}$ is written in reduced form with |P| < |Q|, we will generalize the curious property "$Per_{\beta}(\frac{P}{Q})=Per_{\beta}(\frac{1}{Q})$".

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  1. B. Adamczewski, C. Frougny, A. Siegel, and W. Steiner, Rational numbers with purely periodic beta-expansion, Bull. London Math. Soc. 42 (2010), no. 3, 538-552. https://doi.org/10.1112/blms/bdq019
  2. S. Akiyama, Pisot number and greedy algorithm, Number Theory (Eger, 1996), 9-21, de Gruyter, 1998.
  3. P. Bateman and A. L. Duquette, The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series, Illinois J. Math. 6 (1962), 594-606.
  4. D. W. Boyd, Salem numbers of degree four have periodic expansions, Theorie des nombres (Quebec, PQ, 1987), 57-64, de Gruyter, Berlin, 1989.
  5. D. W. Boyd, On the beta expansion for Salem numbers of degree 6, Mathematics of Computation 65 (1996), no. 214, 861-875. https://doi.org/10.1090/S0025-5718-96-00700-4
  6. R. Ghorbel, M. Hbaib, and S. Zouari, Purely periodic beta-expansions over Laurent series, Internat. J. Algebra Comput. 22 (2012), no. 2, 1-12.
  7. M. Hbaib and M. Mkaouar, Sur le beta-developpement de 1 dans le corps des series formelles, Int. J. Number Theory 2 (2006), no. 3, 365-378. https://doi.org/10.1142/S1793042106000619
  8. S. Ito and H. Rao, Purely periodic ${\beta}$-expansions with Pisot unit base, Proc. Amer. Math. Soc. 133 (2005), no. 4, 953-964. https://doi.org/10.1090/S0002-9939-04-07794-9
  9. B. Li and J. Wu, Beta-expansions and continued fraction expansion over formal Laurent series, Finite Fields Appl. 14 (2008), no. 3, 635-647. https://doi.org/10.1016/j.ffa.2007.09.005
  10. B. Li, J. Wu, and J. Xu, Metric properties and exceptional sets of ${\beta}$-expansions over formal Laurent series, Monatsh. Math. 155 (2008), no. 2, 145-160. https://doi.org/10.1007/s00605-008-0531-7
  11. A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar 8 (1957), 477-493. https://doi.org/10.1007/BF02020331
  12. K. Scheicher, Beta-expansions in algebraic function fields over finite fields, Finite Fields Appl. 13 (2007), no. 2, 394-410. https://doi.org/10.1016/j.ffa.2005.08.008
  13. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), no. 4, 269-278. https://doi.org/10.1112/blms/12.4.269