Do there exist real numbers whose integer powers are bounded away from integers? More precisely, for an arbitrary constant 0 < $\epsilon$ < 1/2, does there exist real x such that for all positive integer n,  

$ \epsilon < (x^n)$ mod 1 < $1-\epsilon $?


Pisot numbers more or less provide the opposite of this behavior, and at first I briefly thought that applying Hurwitz' Theorem to the logs might help...but ln(3)/ln(2) isn't rational, and $2^n$ and $3^n$ seem to keep a minimum unit distance apart. ^\_^



Reals with integer powers bounded away from integers?
We should be able to construct such a real number $x$. Let $\epsilon\in(0,1/2)$ be fixed. If $S\subset \mathbb{R}\_{>0}$ and $r>0$, we write $S^r$ for the set of positive $r$-th powers of elements of $S$. Recursively define a sequence of intervals $I\_n$ of the form $[N\_n+\epsilon,N\_n+1-\epsilon]$ where the $N\_n$ are positive integers. Let $I\_1=[N\_1+\epsilon,N\_1+1-\epsilon]$ where $N\_1$ chosen large enough that $N\_1(1-2\epsilon)>2$. Once $I\_n$ has been defined, $I\_n^{(n+1)/n}$ has length at least $2$, so we can find an integer $I\_{n+1}$ so that $[I\_{n+1}+\epsilon,I\_{n+1}-\epsilon]\subset I\_n^{(n+1)/n}$. Then the intervals $I\_n^{1/n}$ are nested and have diameter going to 0, and we choose $x$ to be the unique element in the intersection of $I\_n^{1/n}$. This $x$ is constructed in such a way that $x^n\in[I\_n+\epsilon,I\_n+1-\epsilon]$ for all $n$.


