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For which $\alpha \in \mathbb R$ can one say that $\forall \epsilon > 0$ there $\exists N \geq 1$ such that $\forall i \in \mathbb N$ one has that some $n\in \{1, \dots, N\}$ is a solution to

$$ \min_{m\in \mathbb Z} |\alpha^{in} - m| \leq \epsilon. $$

Can one find such a bound $N$ for all real numbers $\alpha$? Can someone see any sufficient conditions on $\alpha$ such that this works out? What about $\alpha = e$, that is $\alpha^{in} = \exp(in)$?

Thanks

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up vote 1 down vote accepted

It's true for any algebraic integer $\alpha$ having exactly one conjugate outside the unit circle, since for such $\alpha$ the distance from $\alpha^n$ to the nearest integer goes to zero as $n$ goes to infinity. I suspect it's a very hard question for other real $\alpha$ --- it's a notoriously hard problem to say anything much about small values of the fractional part of $(3/2)^n$, for example, a question of Mahler related to Waring's problem (Mahler's paper is here).

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(The link does not seem to work.) –  Andres Caicedo Jan 21 '13 at 0:56
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@Andres and all, sorry. The bibliographic reference is K Mahler, An unsolved problem on the powers of $3/2$, J Austral Math Soc 8 (1968) 313-321. Maybe one needs a subscription. Maybe journals.cambridge.org/abstract_S1446788700005371 works better --- it may link to the abstract, from which there seems to be a link to the full article. If that doesn't work, try the Austral Math Soc website. –  Gerry Myerson Jan 21 '13 at 1:35
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