For $r\in\mathbb{R}$, define $\Vert{r}\Vert:=\inf_{n\in\mathbb{Z}}\vert r-n\vert$. By Question 4208947 and Question 1536761, we know that $\lim_{n\rightarrow \infty}\Vert(3+2\sqrt{2})^n\Vert=0$ and $\lim_{n\rightarrow \infty}\Vert(2+\sqrt{3})^n\Vert=0$. The method is interesting but can only be applied to restricted numbers. I am wondering whether $\lim_{n\rightarrow \infty}\Vert(3+\sqrt{2})^n\Vert$ exists and is there $s\in \mathbb{R}\setminus\{0\}$ such that $\lim_{n\rightarrow \infty}\Vert(3+\sqrt{2})^n\cdot s\Vert=0$.
Any advice would be helpful. Thanks!
Here is an obligatory graph answer that does not fit in the comments. Link to Sage for you to test yourself! It plots $y = \{ (3 + \sqrt{2})^n \}$. As expected, it behaves randomly. I wonder if something said about its pseudorandomness i.e. it's distributed evenly across $[0, 1)$, but it seems difficult.