Cut the unit circle $S^1=\mathbb{R}/\mathbb{Z}$ at the points
\{\sqrt{1}\}}, \{\sqrt{2}\},\ldots,\{\sqrt{N}\}, where $\{x\} =
x \bmod 1$, and let $J_1, \ldots, J_N$ denote the complementary
intervals, or \emph{gaps}, that remain. We show that, in contrast
to the case of random points (whose gaps are exponentially
distributed), the lengths $|J_i|/N$ are governed by an explicit
piecewise real-analytic distribution $F(t) \,dt$ with phase
transitions at $t=1/2$ and $t=2$.
¶ The gap distribution is related to the probability $p(t)$ that a
random unimodular lattice translate $\Lambda \subset \mathbb{R}^2$
meets a fixed triangle $S_t$ of area $t$; in fact, $p''(t) =
-F(t)$. The proof uses ergodic theory on the universal elliptic
curve
¶ \[
E = \big(\SL_2(\mathbb{R}) \ltimes \mathbb{R}^2\big)/
\big(\SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\big)
\]
¶
and Ratner's theorem on unipotent invariant measures.