Gaps in √n mod 1 and ergodic theory

By Noam D. Elkies, Curtis T. McMullen. 2004

109 citations recorded by the source corpus.

Cut the unit circle S = 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 ⊂ R 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)⋉R 2)/(SL2(Z)⋉ Z ) and Ratner’s theorem on unipotent invariant measures.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/43c73e328ab511ae4d0b89d29bd827a79dced661

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In Geometry and Topology

1. Jens Marklof
2. Andreas Strombergsson
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Citations in the corpus

• DISTRIBUTION MODULO ONE AND RATNER’S THEOREM (2006, methodology): “In these lectures we discuss a few simple applications of one of the central results in measure rigidity: Ratner’s theorem.”
• The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems (2007, background): “…λ({v ∈ Sd−11 : Nc,T (Q,v) = r}) exists, and is given by (3.24) (λ × µ1)({(v,M) ∈ Sd−11 ×X1 : #(Zd∗M ∩ Z(c,Q,v)) = r}) if α ∈ Zd (λ × µq)({(v,M) ∈ Sd−11 ×Xq : #((Zd + pq )M ∩ Z(c,Q,v)) = r}) if α = p q ∈ Qd \ Zd (λ × µ)({(v, g) ∈ Sd−11 ×X : #(Zdg ∩ Z(c,Q,v)) = r}) if α /∈ Qd, where (3.25)…”