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Pisot numbers; fractional parts; limit points
Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\lbrace \lambda \theta ^{n}\rbrace $, $n=0,1,2, \dots $, where $\theta $ is a Pisot number and $\lambda \in \mathbb{Q}(\theta )$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence $(\lambda _{n})_{n\ge 0}$ of elements of $\mathbb{Q}(\theta )$ such that $\operatorname{Card}\,(L(\theta ,\lambda _{n}))< \operatorname{Card}\,(L(\theta ,\lambda _{n+1}))$, $\forall $ $n\ge 0$. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval.
[1] Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugo, M., Pathiaux-Delefosse, M., Schreiber, J.P.: Pisot and Salem numbers. Birkhäuser Verlag Basel, 1992. MR 1187044
[2] Boyd, D.W.: Linear recurrence relations for some generalized Pisot sequences. Advances in Number Theory, Proc. of the 1991 CNTA Conference, Oxford University Press, 1993, pp. 333–340. MR 1368431 | Zbl 0790.11012
[3] Bugeaud, Y.: An introduction to diophantine approximation. Cambridge University Press, Cambridge, 2012. MR 2953186
[4] Dubickas, A.: On the limit points of the fractional parts of powers of Pisot numbers. Arch. Math. (Brno) 42 (2006), 151–158. MR 2240352 | Zbl 1164.11026
[5] Marcus, D.: Number fields. Springer, Berlin, 1977, 3rd edition. MR 0457396 | Zbl 0383.12001
[6] Smyth, C.J.: The conjugates of algebraic integers. Amer. Math. Monthly 82 (86) (1975).
[7] Zaïmi, T.: Comments on the distribution modulo one of powers of Pisot and Salem numbers. Publ. Math. Debrecen 80 (2012), 417–426. DOI 10.5486/PMD.2012.5098 | MR 2943014 | Zbl 1263.11067
[8] Zaïmi, T.: On the spectra of Pisot numbers. Glasgow Math. J. 54 (2012), 127–132. DOI 10.1017/S0017089511000462 | MR 2862390 | Zbl 1303.11118
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