Central limit theorems for generic lattice point counting
Consider the space \(X\) of unimodular lattices in \(\mathbb{R}^d\) with \(d \ge 9\), equipped with the unique \(\mathrm{SL}_d(\mathbb{R})\) invariant measure \(\mu\). The authors prove a central limit theorem for the discrepancy function for the number of points from a generic lattice in a region defined by linear forms. More precisely, for \(L_1, \dots, L_d :\mathbb{R}^d \rightarrow\mathbb{R}\) being linearly independent linear forms, define \(N(x) = L_1(x) \cdots L_d(x)\). Now, for \(I\) a bounded interval in \(\mathbb{R}^+\) and \(T > 0\), define the region \[ \Omega_T = \Omega_T(I) = \left\{x \in\mathbb{R}^d : N(x) \in I\text{ and } 0 < L_1(x), \dots, L_d(x) < T\right\}. \] For a lattice \(\Lambda\), The number of lattice points in \(\Omega_T\) is expected to grow like the volume of \(\Omega_T\) divided by the volume of the fundamental region of \(\Lambda\), so it is natural to consider the discrepancy, \[ \mathcal{D}_T(\Lambda) = \vert \Lambda \cap \Omega_T \vert -\frac{\mathrm{Vol}(\Omega_T)}{\mathrm{Vol}(\mathbb{R}^d/\Lambda)}. \] In the paper under review, the authors show that for \(\xi \in\mathbb{R}\), \[ \mu(\{\Lambda \in X:\mathrm{Vol}(\Omega_T)^{-1/2}\mathcal{D}_T(\Lambda) < \xi\}) \rightarrow \frac{1}{\sigma(I)\sqrt{2\pi}}\int_{-\infty}^\xi e^{-t^2/2\sigma(I)^2}dt, \] as \(T \rightarrow \infty\), where the variance \(\sigma(I)^2\) is explicitly given in terms of the interval \(I\). In fact, a somewhat more general central limit theorem is derived. The proof depends on a counting technique of independent interest developed in the paper, using so-called ``functional tilings''. This takes up the bulk of the paper, and the method may be useful in other counting problems.
scientific article; zbMATH DE number 7634428
scientific article