Gaps in √n mod 1 and ergodic theory

Abstract

Cut the unit circle S1 = ℝ/ℤ at the points {√1}, {√2},..., {√N}, where (x) = x mod 1, and let J1,..., JN 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 |Ji|/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 Λ ⊂ ℝ2 meets a fixed triangle St of area t; in fact, p″(t) = -F(t). The proof uses ergodic theory on the universal elliptic curve E = (SL2(ℝ) ⋉ ℝ2)/(SL2(ℤ) ⋉ ℤ2) and Ratner's theorem on unipotent invariant measures.