Central limit theorems for Diophantine approximants (Q2314798)

In the present paper, the authors prove a central limit theorem for a counting function for the number of solutions to certain Diophantine inequalities. Concretely, let \(\Vert \cdot \Vert\) be some norm on \(\mathbb{R}^n\), let \(\vartheta_1, \dots, \vartheta_m > 0\) and let \(w_1, \dots w_m > 0\) with \(w_1 + \cdots + w_m = n\). For an \(m \times n\) matrix \(u\) with entries in \([0,1]\), let \(L_u^{(i)}\) denote the linear form on \(\mathbb{R}^n\) with coefficients from the \(i\)'th row of \(u\) and consider the system of Diophantine inequalities \[ \big\vert p_i + L_u^{(i)}(q_1, \dots q_n)\big\vert < \vartheta_i \Vert q \Vert^{-w_i}, \quad i = 1,\dots, m, \] with \((p,q) = (p_1, \dots, p_m, q_1, \dots, q_n) \in\mathbb{Z}^m \times (\mathbb{Z}^n\setminus \{0\})\). The authors consider the counting function \(\Delta_T(u)\), which counts the number of solutions to this system of inequalities with \(0 < \Vert q \Vert < T\). It is shown that the function \(\Delta_T(u)\) satisfies a central limit theorem. Namely, if \(m \ge 2\), \(C_{m,n} = 2^m \vartheta_1 \cdots \vartheta_m \int_{S^{n-1}} \Vert z \Vert^{-n}dz\) and \(\xi \in\mathbb{R}\), \[ \bigg\vert \bigg\{ u \in M_{m,n}([0,1]):\frac{\Delta_T(u) - C_{m,n}\log T}{(\log T)^{1/2}}< \xi \bigg\}\bigg\vert \longrightarrow \mathrm{Norm}_{\sigma_{m,n}}(\xi), \] where \(\sigma_{m,n} = 2 C_{m,n}(2 \zeta(m+n-1)\zeta(m+n)^{-1}-1)\) with \(\zeta\) denoting the Riemann \(\zeta\)-function, and with \(\mathrm{Norm}_\sigma(\xi)\) denoting the distribution function of the usual normal distribution with variance \(\sigma\), i.e. \[ \mathrm{Norm}_\sigma(\xi) = (2 \pi \sigma)^{-1/2} \int_{-\infty}^{\xi} e^{-s^2/(2\sigma)} ds. \] The main result of the paper generalises an earlier result of \textit{D. Dolgopyat} et al. [J. Éc. Polytech., Math. 4, 1--35 (2017; Zbl 1387.60046)], who proved the result with \(w_i = n/m\) for all \(i = 1, \dots, m\). In the preceding paper, this assumption is critical, and to overcome this obstacle, the authors of the present paper conduct a quantitative study of higher order correlations of functions on spaces of unimodular lattices and develop new methods for estimating cumulants of Siegel transforms.