## Limit points of fractional parts of geometric sequences.(English)Zbl 1249.11066

Let $$\alpha >1$$ be an algebraic number and let $$\xi \neq 0$$ be a real number. The author investigates the limit points of the sequence of fractional parts $$\{\xi \alpha ^n\},$$ $$n=0,1,2,\dots$$. In particular, he sharpens some results of the reviewer [Bull. Lond. Math. Soc. 38, 70–80 (2006; Zbl 1164.11025)] concerning the largest and the smallest limit points of such sequence for some special algebraic numbers $$\alpha$$ and proves some complementary results. For example, let $$\theta =24.69\dots$$ be the unique solution of $$2x^3-50x^2+15x-1=0$$ in the interval $$(1,+\infty)$$. One of his theorems implies that there is a nonzero $$\xi =\xi (\theta)$$ such that $$\{\xi \theta ^n\} <1/34=0.02941\dots$$ for each integer $$n \geq 0$$. On the other hand, the result of the reviewer implies that $$\limsup _{n \to \infty } \{\xi \theta ^n\} \geq 1/51=0.01960\dots$$ for every $$\xi \neq 0$$.
His results can be applied to Pisot numbers $$\alpha$$. Suppose, for example, that $$\alpha >1$$ is a quadratic Pisot number with conjugate $$\alpha '$$ satisfying $$0<\alpha '<2-\sqrt {2}$$. He proves that $\inf _{\xi \notin \mathbb Q(\alpha)} \limsup _{n\to \infty }\{\xi \alpha ^n\}=\frac {1}{\alpha -\alpha _2}$ (see Corollary 2.6) and finds a transcendental number $$\xi$$ for which this infimum is attained.

### MSC:

 11J71 Distribution modulo one

Zbl 1164.11025