Geodesic
Reprint: An unsolved problem on the powers of $3/2$ (1968)
Documenta mathematica, Mahler Selecta (2019), pp. 639-648.

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One says that $\alpha>0$ is a $Z$-number if $0\le \{\alpha (3/2)^n\}1/2$, where $\{x\}$ denotes the fractional part of $x$. In this paper, while not showing existence, Mahler proves that the set of $Z$-numbers is at most countable. More specifically, Mahler shows that, up to $x$, there are at most $x^{0.7} Z$-numbers. \par Reprint of the author's paper [J. Aust. Math. Soc. 8, 313--321 (1968; Zbl 0155.09501)].
Classification : 11-03, 11J54 
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     author = {Mahler, Kurt},
     title = {Reprint: {An} unsolved problem on the powers of \(3/2\) (1968)},
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     volume = {Mahler Selecta},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__S1__a38/}
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Mahler, Kurt. Reprint: An unsolved problem on the powers of \(3/2\) (1968). Documenta mathematica, Mahler Selecta (2019), pp. 639-648. http://geodesic.mathdoc.fr/item/DOCMA_2019__S1__a38/