Evenly Divisible Rational Approximations of Quadratic Irrationalities

Abstract

In a recent paper of Blomer, Bourgain, Radziwiłł 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 $\alpha$.

7 pages