Powers of rational numbers modulo 1 lying in short intervals
Let \(\{x\}\) denote the fractional part of the real number \(x\), and set \(\|x\|=\min(\{x\},1-\{x\})\); that is, \(\|x\|\) is the distance to the nearest integer. We say that a positive real number \(\alpha\) is a \(Z\)--number provided the fractional parts \(\{\alpha(3/2)^n\}\) satisfy \[ 0\leq \{\alpha(3/2)^n\}<\frac{1}{2} \] for all \(n\geq 0\). The existence of \(Z\)--numbers was considered in the 1960s by \textit{K. Mahler} [J. Aust. Math. Soc. 8, 313--321 (1968; Zbl 0155.09501)] and is still open. Partial progress was made by \textit{L. Flatto, J. C. Lagarias} and \textit{A. D. Pollington} [Acta Arith. 70, No. 2, 125--147 (1995; Zbl 0821.11038)], who proved that for positive coprime integers \(p>q>1\) there is no positive real number \(\xi\) such that the fractional parts \(\{\xi(p/q)^n\}\) lie in any open interval of length less than \(1/p\). Specializing the value \(q=2\), a result of \textit{R. Tijdeman} [Norske Vid. Selsk. Skr. 1972, No. 16, 4 p. (1972; Zbl 0227.10025)] implies that there is a nonzero \(\xi\) such that \[ 0< \{\xi(p/2)^n\}<\frac{1}{p-2} \] for all \(n\geq 0\). In this paper, the author proves that for \(q\geq 2\) an integer and \(t\in\{0,\ldots,q-2\}\), there is a positive real number \(\xi\) such that \[ \|\xi(2-1/q)^n-t/(q-1)\|<(q-1)/(2q-1) \] for all \(n\geq 0\). Extending a result of Tijdeman from the above mentioned paper, the author also proves the following result: Let \(p>q>1\) be two coprime integers where \(p\geq 2q+1\), and let \(t\) be an integer satisfying \(0\leq t\leq p-q-1\). Set \(\delta(p,q)=1/q\) for \(q\) odd and \(\delta(p,q)=2/p\) for \(q\) even. Then there is a real number \(\xi>0\) such that \[ \|\xi(p/q)^n-t/(p-q)\|<q(1-\delta(p,q))/2(p-q). \] As a corollary to the first of these results, there exist two positive real numbers \(\xi\) and \(\zeta\) such that \(\|\xi(5/3)^n\|<2/5\) and \(\|\zeta(5/3)^n\|>1/10\) hold for each \(n\geq 0\).
scientific article; zbMATH DE number 5703367
scientific article