Effective Ratner theorem for ASL(2,R) and gaps in \sqrt{n} modulo 1

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 Str\"ombergsson 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 \sqrt{n} modulo 1.