# Article

Full entry | PDF   (0.2 MB)
Keywords:
Pisot numbers; fractional parts; limit points
Summary:
We consider the sequence of fractional parts $\lbrace \xi \alpha ^n\rbrace$, $n=1,2,3,\dots$, where $\alpha >1$ is a Pisot number and $\xi \in {\mathbb Q}(\alpha )$ is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where $\xi =1$ and the unique limit point is zero, was earlier described by the author and Luca, independently.
References:
[1] Bugeaud Y.: Linear mod one transformations and the distribution of fractional parts $\lbrace \xi (p/q)^n\rbrace$. Acta Arith. 114 (2004), 301–311. MR 2101819
[2] Cassels J. W. S.: An introduction to Diophantine approximation. Cambridge University Press, 1957. MR 0087708 | Zbl 0077.04801
[3] Dubickas A.: A note on powers of Pisot numbers. Publ. Math. Debrecen 56 (2000), 141–144. MR 1740499 | Zbl 0999.11035
[4] Dubickas A.: Integer parts of powers of Pisot and Salem numbers. Arch. Math. (Basel) 79 (2002), 252–257. MR 1944949 | Zbl 1004.11059
[5] Dubickas A.: Sequences with infinitely many composite numbers. Analytic and Probabilistic Methods in Number Theory, Palanga 2001 (eds. A. Dubickas, A. Laurinčikas and E. Manstavičius), TEV, Vilnius (2002), 57–60. MR 1964849 | Zbl 1049.11072
[6] Dubickas A.: Arithmetical properties of powers of algebraic numbers. Bull. London Math. Soc. 38 (2006), 70–80. MR 2201605 | Zbl 1164.11025
[7] Flatto L., Lagarias J. C., Pollington A. D.: On the range of fractional parts $\lbrace \xi (p/q)^n\rbrace$. Acta Arith. 70 (1995), 125–147. MR 1322557
[8] Kuba G.: The number of lattice points below a logarithmic curve. Arch. Math. (Basel) 69 (1997), 156–163. MR 1458702 | Zbl 0899.11050
[9] Luca F.: On a question of G. Kuba. Arch. Math. (Basel) 74 (2000), 269–275. MR 1742638 | Zbl 0995.11043
[10] Smyth C. J.: The conjugates of algebraic integers. Amer. Math. Monthly 82 (1975), 86.
[11] Zaimi T.: An arithmetical property of powers of Salem numbers. J. Number Theory (to appear). MR 2256803 | Zbl 1147.11037

Partner of