Universality in the bulk holds close to given points

Let μμ be a measure with compact support. Assume that ξξ is a Lebesgue point of μμ and that μ′μ′ is positive and continuous at ξξ. Let {An}{An} be a sequence of positive numbers with limit ∞∞. We show that one can choose ξn∈[ξ−Ann,ξ+Ann] such that limn→∞Kn(ξn,ξn+aK̃n(ξn,ξn))Kn(ξn,ξn)=sinπaπa, uniformly for aa in compact subsets of the plane. Here KnKn is the nnth reproducing kernel for μμ, and K̃n is its normalized cousin. Thus universality in the bulk holds on a sequence close to ξξ, without having to assume that μμ is a regular measure. Similar results are established for sequences of measures.