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))