Let $z=e^{2\pi i\theta}$ for a fixed real number $\theta$. It's known that if $\theta\not\in\mathbb{Q}$ (is irrational) then the set $S(\theta)=\{z^n: n\in\mathbb{N}\}$ is dense on the unit circle $\Gamma$.


I am (almost) sure the following is true and really "cute", but I couldn't remember a reference.
Can you help pointing to some resources or even confirming the correct formulation?



> 
> **Statement.** For any fixed $N\in\mathbb{N}$ and fixed $\theta\in\mathbb{R}$, the set of distances between neighborly (adjacent) points of the set 
>  $$S\_N(\theta)=\{z^n: 1\leq n\leq N\}\subset\Gamma$$ 
>  has cardinality at most $3$.
> 
> 
> 


**NOTE.** If $\theta\in\mathbb{Q}$, this is obvious because such a cardinality is just $1$.



distance-set along the orbit of $e^{2\pi i\theta}$
Tony van Ravenstein, The three gap theorem (Steinhaus conjecture), J Austral Math Soc 45 (1988) 360-370, available at <https://oeis.org/A000045/a000045_1.pdf>, references several proofs; perhaps the first was V. T. Sós, On the theory of diophantine approximations. I, Acta Math Acad Sci Hungar 8 (1957) 461-472. 


