Mahler's 3/2 problem

In mathematics, Mahler's 3/2 problem concerns the existence of "Template:Math-numbers".

A Template:Math-number is a real number Template:Math such that the fractional parts of

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

are less than Template:Math for all positive integers Template:Math. Kurt Mahler conjectured in 1968 that there are no Template:Math-numbers.

More generally, for a real number Template:Math, define Template:Math as

$\mathrm{\Omega }(\alpha )=\underset{\theta >0}{\inf }\left(\underset{n\to \mathrm{\infty }}{\mathrm{lim\; sup}}\left\{\theta {\alpha }^n\right\}-\underset{n\to \mathrm{\infty }}{\mathrm{lim\; inf}}\left\{\theta {\alpha }^n\right\}\right).$

Mahler's conjecture would thus imply that Template:Math exceeds Template:Math. Flatto, Lagarias, and Pollington showed that

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

for rational Template:Math in lowest terms.

References

• Template:Cite book