Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show that the image of $f$ divides the circle into arcs of $1$, $2$, or $3$ different lengths.
2026-03-26 18:41:06.1774550466
Three-gap problem, easy version.
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