Evenly divisible rational approximations of quadratic irrationalities

摘要

In a recent paper of Blomer, Bourgain, Radziwi and Rudnick [1], the authors proved the existence of small gaps between eigenvalues of the Laplacian in a rectangular billiard with sides pi and , 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.