Gaps in \(\sqrt{n}mod 1\) and Ergodic Theory

Abstract

Cut the unit circle \(S^1 = \mathbb{R}/\mathbb{Z}\) at the points \(\{\sqrt{1}\}, \{\sqrt{2}\}, . . ., \{\sqrt{N}\}\), where \(\{x\} = x mod 1\), and let \(J_1, . . . , J_N\) denote the complementary intervals, or gaps, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths \(\mid J_i\mid/N\) are governed by an explicit piecewise real-analytic distribution \(F(t)dt\) with phase transitions at \(t=\frac{1}{2}\) 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: \(E = (SL_2(\mathbb{R}) ⋉ \mathbb{R}^2) / (SL_2(\mathbb{Z}) ⋉ \mathbb{Z}^2)\)