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 parts, and also a entire number. $$ \left\lbrace x (3/2)^ n \right\rbrace $$ 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