Sine kernel asymptotics for a class of singular measures

We construct a family of measures on RR that are purely singular with respect to the Lebesgue measure, and yet exhibit universal sine kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel–Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.