# Effective Ratner Theorem for ASL(2,R) and gaps in n^(1/2) modulo 1

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Ilya Vinogradov , University of Bristol
Fine Hall 601

Let G=SL(2, R) \ltimes R^2 and \Gamma= SL(2, Z) ltimes Z^2. We prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of \Gamma \quot 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 mod 1.  Joint work with Tim Browning.