$\pi$th roots of unity and clustering on the unit circle
I came across this interesting blog post that posits a curious hypothesis and I would like to share my understanding of it here.
First, my understanding is that for complex $a$, $1^a$ is multi-valued, given by$$1^a=e^{2i \pi n a }, n\in \mathbb{Z}$$
and for irrational $a$ this leads to countably infinite values of the $a$th power of unity.
Consider as a special case the $\pi$th roots of unity ($a=1/\pi$):
$$1^{1/\pi}=e^{2i n}=\cos 2n +i\sin 2n, $$
and for positive integer $N$, consider the set
$$P_N\equiv \bigcup_{n=0}^{N-1}\{\cos 2n+i\sin 2n\}.$$
An interesting pattern appears to emerge when one plots $P_N$ on the unit circle for various $N$:
$P_3:\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad P_4$:
$P_{22}:\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad P_{23}$:
$P_{355}:\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad P_{350}$:
Note how the points in $P_N$ appear to be equally spaced on the unit circle for certain special $N$ (e.g. $N=3,22,355$) but form clusters for other $N$ (highlighted are clear departures from equal spacing).
The significance of the special $N$ is that they are numerators of the convergents obtained from the continued fraction representation of $\pi$:
$$\pi =[3;7,15,1,292,...],\\ [3]=\frac{3}{1},[3;7]=\frac{22}{7},[3;7,15]=\frac{333}{106},[3;7,15,1]=\frac{355}{113}.$$
It seems not all convergents correspond with equal spacing:
$P_{333}$:
So to summarize my questions:
- If the points in $P_N$ are almost* equally spaced, is $N$ a numerator of a convergent of $\pi$? Or is this simply an embarrassing case of mathematical apophenia?
- If 1. is true, why? And why do some convergents not correspond with equal spacing?
You can obtain the plots above by this Desmos calculator (for large $N$, you may have to zoom in quite a bit to see unequal spacing).
*Update: As the comments point out, they are not exactly equally spaced. For three points, equal spacing would correspond with angles $0,2\pi/3,4\pi/3,$ but these are rather close to $0,2,4$ (i.e. $2n$ for $n=0,1,2$). Is there a justification behind these approximants, somewhat analogous to almost integers?
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