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

Abstract

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.

Reprint of the author's paper [J. Aust. Math. Soc. 8, 313--321 (1968; Zbl 0155.09501)].