A Central Limit Theorem for Counting Functions Related to Symplectic Lattices and Bounded Sets

Abstract

We use a method developed by Björklund and Gorodnik to show a central limit theorem (as $T$ tends to $\infty$) for the counting functions $\#\left(\Lambda \cap {\Omega }_T\right)$ where $\Lambda$ ranges over the space $Y_{2d}$ of symplectic lattices in ${\mathbb{R}}^{2d}$ ($d⩾4$). Here $\{{\Omega }_T{\}}_T$ is a certain family of bounded domains in ${\mathbb{R}}^{2d}$ that can be tessellated by means of the action of a diagonal semigroup contained in $\mathrm{Sp}(2d,\mathbb{R})$. In the process we obtain new $L^p$ bounds on a certain height function on $Y_{2d}$ originally introduced by Schmidt.

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