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.


11J71 Distribution modulo one


Zbl 1164.11025