On chaotic and random sequences
\textit{J .J. Suárez, I. Rondón, L. Trujillo} and \textit{J. A. Gonzáles} [Chaos Solitons Fractals 21, No. 3, 603--622 (2004; Zbl 1048.37516)] proved, for the sequence \(X_n=\sin^2(\theta\pi Z^n)\), \(n=0,1,2,\dots\), where \(Z\) is a rational number expressed as \(Z=\frac{p}{q}\), that from the observation of the previous values \(X_s,X_{s+1},\dots,X_{s+m}\) there is no method for determining the next value \(X_{s+m+1}\). From this they deduce that \(X_n\) is a good candidate for randomness and they affirm that \(Y_n=\frac{2}{\pi}\arcsin(\sqrt X_n)\) is a uniformly distributed sequence. Note that the distribution of the fractional part of \((p/q)^n\bmod1\) is largely unsolved problem (for the current state of stage, see the reviewer and \textit{Š. Porubský} [Distribution of Sequences: A Sampler, Peter Lang, Frankfurt am Main (2005; Zbl 1078.11051), p.~2--149, 2.17]). The following is only known: (1) \(\limsup_{n\to\infty}\theta(p/q)^n\bmod1- \liminf_{n\to\infty}\theta(p/q)^n\bmod1\geq1/p\) for \(\theta>0\) (see \textit{L. Flatto, J. C. Lagarias} and \textit{A. D. Pollington} [Acta Arith. 70, No. 2, 125--147 (1995; Zbl 0821.11038)]). (2) No explicit example of a real number \(Z\) is known for which the sequence \(Z^n\bmod1\), \(n=1,2,\dots\), is uniformly distributed. (3) Let \(Z>1\) be an algebraic integer such that all the conjugates of \(Z\) have modulus \(<1\) (called Pisot-Vijayaraghavan number), then \(\lim_{n\to\infty}Z^n\bmod1=0\). (4) If \(\theta\) is a normal number in the integer base \(q\), then the sequence \(\theta q^n\bmod1\), \(n=1,2,\dots\) is uniformly distributed in \([0,1]\). In the paper under review, the author gives a detailed but elementary proof that \[ \lim_{n\to\infty}\sin^2(\pi Z^n)=0 \] for a Pisot-Vijayaraghavan number \(Z=m+\sqrt{m^2-1}\), where \(m\) is an integer. From this proof he deduces that \(\lim_{n\to\infty}\sin^2(\frac{1}{2}\pi Z^n)=0\) for every integer \(m\) and \(\lim_{n\to\infty}\sin^2(\frac{1}{4}\pi Z^n)=0\) for \(m=2\).
scientific article; zbMATH DE number 2132815
scientific article