Evenly divisible rational approximations of quadratic irrationalities (Q1709744)

In [\textit{V. Blomer} et al., Ann. Sci. Éc. Norm. Supér. (4) 50, No. 5, 1283--1300 (2017; Zbl 1405.35120)], the authors investigated the minimal gaps between energy levels of the eigenvalues of the Laplacian of a rectangular billiard with width \(\pi/\sqrt{\alpha}\) and height \(\pi\). These eigenvalues are of the form \(\alpha m^2+n^2\), where \(m,n\geq 1\) are integers. If \(\alpha\) is a positive irrational, these energy levels belong to a simple spectrum \(0<\lambda_1<\lambda_2< \cdots\), with growth \(\lambda_N \sim \frac{4\sqrt{\alpha}}{\pi}N \). The minimal gap function is then defined as \[ \delta_{\min}^{(\alpha)}(N): = \min(\{\lambda_{i+1}-\lambda_{i}: 1\leq i < N\}). \] The author of this paper generalizes the results of Blomer et al. [loc. cit.] to all positive quadratic irrationalities \(\alpha\).