Gaps in √n mod 1 and ergodic theory

Abstract

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 \[ E = \big(\mathrm{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2\big)/ \big(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\big) \] and Ratner's theorem on unipotent invariant measures.