It is known that if you choose n point at random on S^{1} = [0,1], the nearest neighbor spacings between the points are exponentially distributed with mean 1.

For example, two of our n points could be x, x + α/n Then with probability (1 - α/n)^{n-2} -> e^{-α} The remaining n-2 points avoid the interval [x, x + α/n].

For the sequence kα (mod 1) does the nearest neighbor spacings of the first n elements approach a limit? I've heard it does when α is the golden ratio $(\sqrt{5}-1)/2$, but not in general. So maybe this is only true for specific α irrational.

Any readings on this problem would be appreciated, too.

**EDIT**: Apparently this sequence arises in the energy spacings of the two dimensional quantum harmonic oscillator. The equation would be $ - \frac{\hbar^2}{2m}\left( \frac{d^2 \psi}{dx^2} + \frac{d^2 \psi}{dx^2} \right) + \frac{1}{2}m[(\omega_1 x)^2 + (\omega_2y)^2]= E \psi$ and the spectrum would be $\frac{\hbar}{2}(m \omega_1 + n \omega_2 )$. This comes from Pavel Bleher's The energy level spacing for two harmonic oscillators with golden mean ratio of frequencies