Evenly Divisible Rational Approximations of Quadratic Irrationalities

TL;DR

This paper extends previous results on small gaps between eigenvalues of the Laplacian in rectangular billiards to all positive quadratic irrationalities, providing a broader understanding of their spectral properties.

Contribution

It generalizes prior work by proving the existence of small gaps for all positive quadratic irrationalities, not just specific types.

Findings

Small gaps exist for all positive quadratic irrationalities

Extension of spectral gap results to a wider class of irrationalities

Broader implications for eigenvalue distribution in billiards

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 $\alpha$.