Mahler's 3/2 problem

In mathematics, Mahler's 3/2 problem concerns the existence of " $Z$ -numbers".

A $Z$ -number is a real number $x$ such that the fractional parts of

x\left({\frac {3}{2}}\right)^{n}

are less than $1/2$ for all positive integers $n$ . Kurt Mahler conjectured in 1968 that there are no $Z$ -numbers.

More generally, for a real number $α$ , define $Ω(α)$ as

\Omega (\alpha )=\inf _{\theta >0}\left({\limsup _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace -\liminf _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace }\right).

Mahler's conjecture would thus imply that $Ω(3/2)$ exceeds $1/2$ . Flatto, Lagarias, and Pollington showed^[1] that

\Omega \left({\frac {p}{q}}\right)>{\frac {1}{p}}

for rational $p / q > 1$ in lowest terms.

References

Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. ISBN 0-8218-3387-1. Zbl 1033.11006.