Suppose $\alpha >1$ such that the distance from $\alpha^n$ to the integers tends to zero as $n$ tends to infinity.


Question 1: Can $\alpha$ be a rational number?


Question 2: The property is satisfied if $\alpha$ is a root of a monic polynomial with integer coefficient that has all the other roots with module less than 1. Is the converse true?



On the number $\alpha>1$ such that $lim_{n \to \infty}{d(\alpha^n, \mathbb{Z})}$ = 0
1) No, it can not unless it is integer.
2) This is an open problem known as Pisot-Vijayaraghavan problem. For algebraic numbers the answer is positive, that implies the answer to 1). 
See 
<https://en.m.wikipedia.org/wiki/Pisot-Vijayaraghavan_number>


Short self-contained answer to 1): if $\alpha^n=A\_n+\delta\_n$ for positive integer $A\_n$ and $\delta\_n\to 0$, and $\alpha=p/q$, we get $qA\_{n+1}-pA\_n=-q\delta\_{n+1}+p\delta\_n\to 0$, but $qA\_{n+1}-pA\_n$ is integer, hence for large enough $n$ we have $A\_{n+1}=\frac pq A\_n$, this is impossible since such a sequence takes non-integer values for coprime $p,q>1$.


