Mahler's $3/2$ problem described here. Am I right in thinking the following would prove it?
Show that for every integer $m$, there is some integer $n$ for which the fractional part $$\left\{\left(\frac{3}{2}\right)^n\right\}\gt1-\frac{1}{2^m}$$
Because for every real number $x$ which generates some fractional part there will be an $x=1/2^m$ which generates a still smaller frational part.
If so, the problem does not look so difficult on the face of it. In fact surely the irrationality of $\log_2{(3)}$ ensures it?
If my restatement is not correct, is there an example of when and why my restatement would not correspond with the original form?