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Let $[a]$ denote the integral part of a real number $a$. Let $a$ be an irrational number and $b$ a real number greater than $1$. Consider the sequence $(b^n(na-[na]))$ with $n$ running on the positive integers. I would like to know if this sequence diverges. In other words I want to know if the inverse of the sequence of truncations $(na-[na])$ growths slower than any exponentional sequence.

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    $\begingroup$ From equidistribution $na - [na] > \frac12$ infinitely frequently. Thus for any function $f:\mathbb{N}\to\mathbb{R}$ with $\lim_{n\to\infty} f(n) = \infty$ the sequence $f(n) (na - [na])$ cannot remain bounded. But I think this is not exactly the question you asked in the final sentence. The final sentence seems to be asking about the $\liminf$ of $b^n (na - [na])$. Please edit your question to clarify. $\endgroup$
    – Willie Wong
    Commented May 27, 2014 at 14:59

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If, as Willie Wong suggests, the question is about the lim inf of $b^n (n a - \lfloor n a \rfloor)$, that is almost always $+\infty$ but generically $0$ (i.e. there is a dense $G_\delta$ set of $a$'s on which it is $0$).

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