It is an open problem whether the fractional part of $\left(\dfrac32\right)^n$ is dense in $[0...1]$.

The problem is: is $1$ a limit point of the above sequence?

An equivalent formulation is: $\forall \epsilon > 0: \exists n \in \Bbb N: 1 - \{1.5^n\} < \epsilon$ where $\{x\}$ denotes the fractional part of $x$.

Here is a table of $n$ against $\epsilon$ that I computed:

$\begin{array}{|c|c|}\hline \epsilon & n \\\hline 1 & 1 \\\hline 0.5 & 5 \\\hline 0.4 & 8 \\\hline 0.35 & 10 \\\hline 0.3 & 12 \\\hline 0.1 & 14 \\\hline 0.05 & 46 \\\hline 0.01 & 157 \\\hline 0.005 & 163 \\\hline 0.001 & 1256 \\\hline 0.0005 & 2677 \\\hline 0.0001 & 8093 \\\hline 0.00001 & 49304 \\\hline 0.000005 & 158643 \\\hline 0.0000005 & 835999 \\\hline \end{array}$

References

1. Unsolved Problems, edited by O. Strauch, in section 2.4 Exponential sequences it is explicitly mentioned that both questions whether $(3/2)^n\bmod 1$ is dense in $[0,1]$ and whether it is uniformly distributed in $[0,1]$ are open conjectures.
2. Power Fractional Parts, on Wolfram Mathworld, "just because the Internet says so"

Another comment, but too big for the standard box. An atanh() rescaling might be an interesting thing, see my example:

The pink and the blue lines are hullcurves connecting the points $\small (N,f(N))$ where $f(N)$ is extremal (with moving maxima/minima) and the grey dots are points $\small (N,f(N))$ at $\small N \le 1000 $ which shall illustrate the general random distribution of the $\small f(N)$.
The grey lines are manually taken smooth subsets of the extremaldata and symmetrized (by merging of datasets and adapting sign) to show the rough tendency of extension of the vertical intervals.

I liked it that the atanh()-scaling seem to suggest some roughly linear increase/decrease of the hullcurves.

[update] The data for the picture were extended by data from the OP and OEIS A153663 (magenta upper curve) and from the OEIS A081464 (blue lower curve). Note, that the OEIS has even more datapoints, but that needed excessive memory/time to compute the high powers of (3/2) and its fractional parts.