Particular questions such as [Why is $\pi$ so close to $3$?](https://math.stackexchange.com/questions/729907/why-is-pi-so-close-to-3) or [Why is $\pi^2$ so close to $10$?](https://math.stackexchange.com/questions/1644136/why-is-pi2-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 [$\pi^3>31$](https://math.stackexchange.com/a/1651175/134791) 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:

1. 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 $\frac{1}{4}$?
2. Let $\lfloor x \rceil$ denote the [rounding function](http://functions.wolfram.com/IntegerFunctions/Round/introductions/FloorRelated/ShowAll.html). 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](https://math.stackexchange.com/questions/1634065/is-there-a-section-of-mathematics-that-studies-near-integer-equations?lq=1))

In either case:

> 
> **Q**: Is the difference between $\pi^k$ and its nearest integer [uniformly distributed](http://mathworld.wolfram.com/UniformDistribution.html) in $(-\frac{1}{2},\frac{1}{2})$?
> 
> 
>