Powers of rational numbers modulo 1 lying in short intervals
type: Item
instance of: scholarly article
title: Powers of rational numbers modulo 1 lying in short intervals
zbMATH Open document ID: 1205.11085
published in: Results in Mathematics
publication date: 2010-05-03T00:00:00Z
review text: 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 \(n\geq 0\).
reviewed by: Michael Coons
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11J54
zbMATH DE Number: 5703367
zbMATH Keywords: Fractional part
zbMATH Keywords: Mahler Z-number
zbMATH Keywords: integer sequence
zbMATH Keywords: Abel's lemma
author: A. Dubickas
MaRDI profile type: Publication
full work available at URL: s00025-009-0001-0
OpenAlex ID: W2020142790
cites work: On the chromatic number of a distance graph associated with a lacunary sequence
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cites work: Linear mod one transformations and the distribution of fractional parts {ξ(p/q)n}
cites work: On the fractional parts of lacunary sequences
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cites work: On the powers of 3/2 and other rational numbers
cites work: An Approximation by Lacunary Sequence of Vectors
cites work: Powers of a rational number modulo 1 cannot lie in a small interval
cites work: On the range of fractional parts {ξ(p/q)ⁿ}
cites work: Distribution of geometric sequences modulo 1
cites work: An unsolved problem on the powers of 3/2
cites work: Q5634051
DOI: 10.1007/S00025-009-0001-0
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description: scientific article; zbMATH DE number 5703367