TITLE: Is $\pi^k$ any closer to its nearest integer than expected?
QUESTION [13 upvotes]: Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest integer? 
For instance, we may stare in awe in front of the almost-unit 
$$\frac{\pi}{31^\frac{1}{3}}=1.000067...$$
or, in binary system, 
$$\frac{\pi}{11111_{2}^\frac{1}{11_2}} \approx 1$$
so proving that $\pi^3>31$ becomes interesting, but it would not be striking that sometimes $\pi^k$ lied close to its nearest integer if that was balanced by other unlucky times when it would be at almost half a unit, by the straightforward effect of the rounding function.
Under a uniform distribution assumption, the expected distance between $\pi^k$ and its nearest integer is $\frac{1}{4}$, and the title has at least the following two interpretations, as a random variable (given this made sense) and as particular outcomes of a random variable:

In average, is $\pi^k$ any closer to its nearest integer than expected? The first three powers have differences less than $\frac{1}{4}$ in absolute value, namely 
$$\begin{align}
\pi-3 
&\approx 
.1416 \\
\pi^2-10 
&\approx 
-0.1304 \\ 
\pi^3-31 
&\approx0.0063
\end{align}$$
This event has probability $\left(\frac{1}{2}\right)^3=\frac{1}{8}$ assuming independence. What happens as the number of powers considered grows? Another approach, with median instead of average: Does the median of the absolute value of that difference tend to $\frac{1}{4}$?
Let $\lfloor x \rceil$ denote the rounding function. Although the answer to the first question may be false, for what values of $k$ does $\lfloor x^k \rceil ^\frac{1}{k}$ yield more bits of $\pi$ than it uses? For instance, $\lfloor x^{157} \rceil ^\frac{1}{157}$ seems to be an interesting approximation to $\pi$. (See this question)

In either case:

Q: Is the difference between $\pi^k$ and its nearest integer  uniformly distributed in $(-\frac{1}{2},\frac{1}{2})$?

REPLY [8 votes]: It's mentioned here that the sequence $x^n$ ($n=1,2 \cdots$) in modulo $1$ is known to be  uniformly distributed  for almost every $x>1$. At the same time, and perhaps surprisingly, not even a single example has been discovered - only some exceptions (and all algebraic). Furthermore, it has been proved (see same reference) that the "exceptions", in spite of having Lebesgue measure zero, are uncountable (hence they must include trascendental numbers). 
I didn't find anything about the particular case $x=\pi$, neither about what happens with the distribution of the fractional parts (do they concentrate around $0$?) in the non-uniform exceptional cases.