  We construct a family of measures on $\bbR$ that are purely singular with
respect to 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.
