Mahler, Kurt

Reprint: An unsolved problem on the powers of \(3/2\) (1968)

Doc. Math. Extra Vol. Mahler Selecta, 639-648 (2019)
DOI: 10.25537/dm.2019.SB-639-648

Summary

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)].

Mathematics Subject Classification

11-03, 11J54

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