View source for Z-number


''Mahler's 3/2 problem'' concerns the existence of "Z-numbers".  A ''Z-number'' is a real number $x$ such that the  [[Fractional part of a number|fractional part]]s
$$
\left\lbrace x (3/2)^ n \right\rbrace
$$
are less than 1/2 for all natural numbers $n$.  Kurt Mahler conjectured in 1968 that there are no Z-numbers.

More generally, for a real number $\alpha$, define $\Omega(\alpha)$ as
$$
\Omega(\alpha) = \inf_\theta\left({ \limsup_{n \rightarrow \infty} \left\lbrace{\theta\alpha^n}\right\rbrace - \liminf_{n \rightarrow \infty} \left\lbrace{\theta\alpha^n}\right\rbrace }\right) \ . 
$$
Mahler's conjecture would thus imply that $\Omega(3/2)$ exceeds 1/2.  This is true, and indeed Flatto, Lagarias and Pollington showed that
$$
\Omega(p/q) > 1/q 
$$
for rational $p/q$.

====References====
* Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas; ''Recurrence sequences'' Mathematical Surveys and Monographs '''104''' American Mathematical Society (2003) ISBN 0-8218-3387-1 {{ZBL|1033.11006}} 

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