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 that


 * $$\Omega\left(\frac p q\right) > \frac 1 p $$

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