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Jul 7, 2017 at 16:14 comment added Yaakov Baruch The scheme above can be used to produce $\alpha$'s whose powers stay away from integers by at least $\epsilon$ for any given $\epsilon<1/2$. Moreover, by picking a big enough starting $\alpha$, the scheme can produce a tree of counterexamples where each node has more than 2 (or more) branches, therefore proving that there are uncountably many such counterexamples.
Jul 5, 2017 at 20:26 history edited Yaakov Baruch CC BY-SA 3.0
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Jul 5, 2017 at 20:03 history answered Yaakov Baruch CC BY-SA 3.0