Can we find a real number $k$ such that the fractional part of sequence $k^n(n\in N)$ monotonically increasing?

Question

Can we find a real number $k$ such that the fractional part of sequence $k^n(n\in N)$ monotonically increasing?

This link may be helpful , it says that Hardy and Littlewood have shown that $frac(k^n)$ are equidistributed over (0,1) for almost all real numbers $k$.

Any ideas or hints will be appreciate.

Answer 1

Let $x=3+2\sqrt{2}\approx 5.8284$; then the fractional part of $x^n$ monotonically increases, converging exponentially fast to $1$.

Other values that work include $\phi^2\approx 2.618$, or the squares of any of the so-called "silver means" mentioned in the link you provided.