Evenly Divisible Rational Approximations of Quadratic Irrationalities

Abstract

In a recent paper of Blomer, Bourgain, Radziwi{\l}{\l} and Rudnick, the authors proved the existence of small gaps between eigenvalues of the Laplacian in a rectangular billiard with sides $\pi$ and $\pi/\sqrt\alpha$, i.e. numbers of the form $\alpha m^2+ n^2$, whenever $\alpha$ is a quadratic irrationality of certain types. In this note, we extend their results to all positive quadratic irrationalities $\alpha$.