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.