If it is just to puzzle the graduate students for 10+ minutes, consider the set of reals $a>1$ such that for some $K>0$ the distance from $Ka^n$ to the nearest integer tends to $0$ as $n\to\infty$. The answer is, indeed, next to obvious but I've seen quite a few graduate students that weren't able to do it off hand. Now take a stopwatch and see how soon you'll solve this one yourself :).
fedja
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