I am looking for real numbers $\ x > 1\ $ such that $\ \{\ \{x^n\}: n\in\mathbb{N}\}\ $ is dense in $[0,1].\ $ Here, $\ \{y\}\ $ means the fractional part of $\ y\in\mathbb{R}.$
What I know:
- I read somewhere on this site that this is known to be true for almost all $\ x\ $ i.e. for all $x\in\mathbb{R}\setminus Y\ $ where $\ Y\ $ is a set of measure zero.
- I know that this is not true for some $\ x\ $ such as $\ x = 1 + \sqrt{2},\ $ because $\ (1+\sqrt{2})^n + (1-\sqrt{2})^n\in\mathbb{Z}.$
- I know that this is an open problem for $\ x=\frac{3}{2}.\ $
But I want to know more about this. For example, is there a classification, as to which real numbers have this density property, for algebraic numbers? Transcendental numbers? And for which other values of $\ x\ $ is this currently an open problem? Are there any good resources on all this?