BETA-EXPANSIONS WITH PISOT BASES OVER Fq((x-1))
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It is well known that if the β-expansion of any nonnegative integer is finite, then β is a Pisot or Salem number. We prove here that $\mathbb{F}_q((x^{-1}))$, the β-expansion of the polynomial part of β is finite if and only if β 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 β is a Pisot series, then there is a bound of the length of the fractional part of β-expansion of any polynomial P in $\mathbb{F}_q[x]$.
Abstract
It is well known that if the β-expansion of any nonnegative integer is finite, then β is a Pisot or Salem number. We prove here that $\mathbb{F}_q((x^{-1}))$, the β-expansion of the polynomial part of β is finite if and only if β 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 β is a Pisot series, then there is a bound of the length of the fractional part of β-expansion of any polynomial P in $\mathbb{F}_q[x]$.
참고문헌 (8)
- S. Akiyama, "Number theory (Eger, 1996)", pp.9~21 (1998), de Gruyter
- P. Bateman, "The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series", Illinois J. Math, 6, pp.594~606 (1962)
- C. Frougny, "Finite beta-expansions", Ergodic Theory Dynam. Systems, 12(4), pp.713~723 (1992)
- M. Hbaib, "Sur le beta-developpement de 1 dans le corps des series formelles", Int. J. Number Theory, 2(3), pp.365~378 (2006)
- M. Hollander, "Linear numeration systems, Finite beta-expansion and discrete spectrum of substitution dynamical systems", 박사, pp.1966
- J. Neukirch, , pp.322~ (1999), Springer-Verlag
- A. Renyi, "Representations for real numbers and their ergodic properties", Acta Math. Acad. Sci. Hungar, 8, pp.477~493 (1957)
- K. Scheicher, "β-expansions in algebraic function fields over finite fields", Finite Fields Appl, 13(2), pp.394~410 (2007)