Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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BETA-EXPANSIONS WITH PISOT BASES OVER Fq((x-1))

  • Hbaib, Mohamed (Departement de Mathematiques Faculte des Sciences de Sfax)
  • Received : 2010.09.01
  • Published : 2012.01.31

Abstract

It is well known that if the ${\beta}$-expansion of any nonnegative integer is finite, then ${\beta}$ is a Pisot or Salem number. We prove here that $\mathbb{F}_q((x^{-1}))$, the ${\beta}$-expansion of the polynomial part of ${\beta}$ is finite if and only if ${\beta}$ is a Pisot series. Consequently we give an other proof of Scheiche theorem about finiteness property in $\mathbb{F}_q((x^{-1}))$. Finally we show that if the base ${\beta}$ is a Pisot series, then there is a bound of the length of the fractional part of ${\beta}$-expansion of any polynomial P in $\mathbb{F}_q[x]$.

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References

  1. S. Akiyama, Pisot numbers and greedy algorithm, Number theory (Eger, 1996), 9-21, deGruyter, Berlin, 1998.
  2. 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.
  3. C. Frougny and B. Solomyak, Finite beta-expansions, Ergodic Theory Dynam. Systems 12 (1992), no. 4, 713-723.
  4. 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
  5. M. Hollander, Linear numeration systems, Finite beta-expansion and discrete spectrum of substitution dynamical systems, Ph. D thesis, University of Washington, 1966.
  6. J. Neukirch, Algebraic Number Theory, Translated from the 1992 German original and with a note by Norbert Schappacher.With a foreword by G. Harder.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322. Springer-Verlag, Berlin, 1999.
  7. 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
  8. 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

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