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In [6], it is proved that the lengths of periods occurring in the β-expansion of a rational series r noted by Perβ(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 β-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if r = P/Q is written in reduced form with |P| < |Q|, we will generalize the curious property “Perβ(P/Q) = Perβ(1/Q)”.
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참고문헌 (13)
참고문헌 신청
A. R`enyi / 1957 / Representations for real numbers and their ergodic properties / Acta Math. Acad. Sci. Hungar 8 : 477 ~ 493
B. Adamczewski / 2010 / Rational numbers with purely periodic beta-expansion / Bull. London Math. Soc 42 (3) : 538 ~ 552
B. Li / 2008 / Beta-expansions and continued fraction expansion over formal Laurent series / Finite Fields Appl 14 (3) : 635 ~ 647
B. Li / 2008 / Metric properties and exceptional sets of β-expansions over formal Laurent series / Monatsh. Math 155 (2) : 145 ~ 160
D. W. Boyd / 1989 / Salem numbers of degree four have periodic expansions, Th´eorie des nom-bres / de Gruyter : 57 ~ 64
B. Adamczewski / 2010 / Rational numbers with purely periodic beta-expansion / Bull. London Math. Soc 42 (3) : 538 ~ 552
B. Li / 2008 / Beta-expansions and continued fraction expansion over formal Laurent series / Finite Fields Appl 14 (3) : 635 ~ 647
B. Li / 2008 / Metric properties and exceptional sets of β-expansions over formal Laurent series / Monatsh. Math 155 (2) : 145 ~ 160
D. W. Boyd / 1989 / Salem numbers of degree four have periodic expansions, Th´eorie des nom-bres / de Gruyter : 57 ~ 64
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