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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.

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    Well, that's not true. See what happens when $a$ is the golden ratio.2011-03-22
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    Here's is the direct link to fedja's post on the MO thread: http://mathoverflow.net/questions/59115/a-set-for-which-it-is-hard-to-determine-whether-or-not-it-is-countable/59130#591302011-03-22
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    You can show that it is countable easy enough (they are all computable numbers, for one thing), but I don't know a simple way of describing the set. Fedja's comment that it is not known if it is a subset of the algebraic numbers suggests nobody else knows either.2011-03-22

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