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.