Effective Ratner Theorem for $ASL(2, \mathbb{R})$ and the gaps of the sequence $\sqrt n$ mod $1$

Let $G=SL(2,\mathbb{R})\ltimes \mathbb{R}^2$ and $\Gamma=SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of $\Gamma\setminus G$, 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 $1$.

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