ON THE PERIOD OF β-EXPANSION OF PISOT OR SALEM SERIES OVER 𝔽q((x-1))

Title & Authors
ON THE PERIOD OF β-EXPANSION OF PISOT OR SALEM SERIES OVER 𝔽q((x-1))
RIM, GHORBEL; SOUROUR, ZOUARI;

Abstract
In [6], it is proved that the lengths of periods occurring in the $\small{{\beta}}$-expansion of a rational series r noted by $\small{Per_{\beta}(r)}$ 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 $\small{{\beta}}$-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if $r Keywords formal power series;$\small{{\beta}}$-expansion;Pisot series;Salem series; Language English Cited by References 1. B. Adamczewski, C. Frougny, A. Siegel, and W. Steiner, Rational numbers with purely periodic beta-expansion, Bull. London Math. Soc. 42 (2010), no. 3, 538-552. 2. S. Akiyama, Pisot number and greedy algorithm, Number Theory (Eger, 1996), 9-21, de Gruyter, 1998. 3. P. Bateman and A. L. Duquette, The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series, Illinois J. Math. 6 (1962), 594-606. 4. D. W. Boyd, Salem numbers of degree four have periodic expansions, Theorie des nombres (Quebec, PQ, 1987), 57-64, de Gruyter, Berlin, 1989. 5. D. W. Boyd, On the beta expansion for Salem numbers of degree 6, Mathematics of Computation 65 (1996), no. 214, 861-875. 6. R. Ghorbel, M. Hbaib, and S. Zouari, Purely periodic beta-expansions over Laurent series, Internat. J. Algebra Comput. 22 (2012), no. 2, 1-12. 7. M. Hbaib and M. Mkaouar, Sur le beta-developpement de 1 dans le corps des series formelles, Int. J. Number Theory 2 (2006), no. 3, 365-378. 8. S. Ito and H. Rao, Purely periodic${\beta}$-expansions with Pisot unit base, Proc. Amer. Math. Soc. 133 (2005), no. 4, 953-964. 9. B. Li and J. Wu, Beta-expansions and continued fraction expansion over formal Laurent series, Finite Fields Appl. 14 (2008), no. 3, 635-647. 10. B. Li, J. Wu, and J. Xu, Metric properties and exceptional sets of${\beta}\$-expansions over formal Laurent series, Monatsh. Math. 155 (2008), no. 2, 145-160.

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