TITLE: The Three Gap Theorem: why should it be true?
QUESTION [14 upvotes]: Fix $\alpha\in\mathbb R$ and $N\in \mathbb N$, consider the set $S(\alpha,N)$ of $\{k\alpha\},k=1,\dots,N$, where $\{x\}$ denotes the fractional part of $x$. Let $a_1,\ldots a_N$ be the elements of $S(\alpha, N)$ arranged in increasing order, and consider the gaps sizes
$$
|a_{i+1}-a_i|, i=1,\dots,N-1.
$$
The well-known Three Gap Theorem tells us that the number of distinct gap sizes is at most three for any $\alpha$ and $N$.
There are many proofs and generalisations of this theorem, but I haven't gained any intuition regarding the statement. In particular, is there any heuristic/intuitive reason as to why the theorem should be true?

REPLY [2 votes]: Its all about modular arithmetic. Draw $n$ points $P_0 = 0, P_1, \ldots, P_{n-1}$, with the largest $P_{max}$, and smallest (non-zero) $P_{min}$, with $min, max \in \{1, \dots, n-1\}$. Now, the initial gap, $a = P_{min} - P_0$, is repeated for every point with at least $min$ successors, i.e. the first $n - min$ points. Similarly, the final gap, $b = P_0 - P_{max}$, is repeated for every point with at least $max$ predecessors, i.e. the final $n - max$ points.
There is also a middle group of $min + max - n$ points/gaps that get bisected into an $a$ and $b$ every time $n$ increments until $n = min + max$, and no more mid-points remain. The next time that $n$ increments, the new point $P_{min + max} = a - b$ bisects the larger of the initial $a$ and final $b$, and it all starts over.
There are usually three gaps, $a$, $b$ and $a + b$, that reduce to two, $a$ and $b$, every time $n = min + max$.