Evenly Divisible Rational Approximations of Quadratic Irrationalities

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 $π$ and $π/\sqrtα$, i.e. numbers of the form $αm^2+ n^2$, whenever $α$ is a quadratic irrationality of certain types. In this note, we extend their results to all positive quadratic irrationalities $α$.

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