Effective Ratner theorem for ASL(2,R) and gaps in √(n) modulo 1

2013/11/25 by Tim Browning, Browning, Tim, Ilya Vinogradov +1
Mathematics · #11J71 37A17 #11L07 #37A25 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #math.DS #math.NT #msc:11J71 #msc:11L07 #msc:37A17 #msc:37A25

paper · pdf · doi:10.48550/arxiv.1311.6387

24 pages; further extensive rewriting of section 5

arxiv created 2016/02/26 · arxiv updated 2016/02/29

Abstract

Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Strömbergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of √(n) modulo 1.

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