I'm referring to an answer posted on Math Overflow (see the post by fedja on https://mathoverflow.net/questions/59115/a-set-for-which-it-is-hard-to-determine-whether-or-not-it-is-countable)
The question is whether the set of real numbers $a > 1$ so that for $K > 0$ the distance between $K a^n$ and its nearest integer approaches $0$ for $n \to \infty$ is countable.
The integers are obviously in that set. However I couldn't come up with a proof that for all other reals the limit does not exist.