Here, $\{ x \}$ denotes the fractional part of $x.$
Are there any known positive integers $k$ for which the set $\left\{\ \left\{ \left( k+ \frac{1}{2} \right)^n \right\}: n\in\mathbb{N}\ \right\} $ is known to be dense in $[0,1]$ ?
I know this is an open problem for $k=1,$ so suspect it might also be an open problem for all other integers $k,$ which then leads to the following question:
If the answer to the first question is no, then is the set $\left\{\ \left\{ \left( k+ \frac{1}{2} \right)^n \right\}: n,k\in\mathbb{N}\ \right\} $ known to be dense in $[0,1]$ ?
For the second question the answer is yes. Note that $$(k + 0.5)^n = 2^{-n} + nk2^{-n+1} + \sum_{j = 2}^{n} \binom{n}{j}k^j 2^{-n+j}.$$ Thus, if we take $n = 2m$, $k = 2^{m - 1} t$ for integer $t$, then for any $j \geq 2$, the number $\binom{n}{j}k^j 2^{-n+j}$ is an integer. So we conclude that $$\{(k + 0.5)^n\} = \{2^{-2m} + mt2^{-m+1}\}.$$ In particular, as $t$ travels from $0$ to $2^{m - 1} / m$, the RHS comes within $m2^{-m+1}$ of any number in $[0,1]$. Taking $m \to \infty$ gives the desired density.