Universality in the bulk holds close to given points

Let be a measure with compact support. Assume that is a Lebesgue point of and that 0 is positive and continuous at : Let fAng be a sequence of positive numbers with limit 1. We show that one can choose n 2 An n ; + An n such that lim n!1 Kn n; n + a ~ Kn( n; n) Kn ( n; n ) = sin a a ; uniformly for a in compact subsets of the plane. Here Kn is the nth reproducing kernel for , and ~ Kn is its normalized cousin. Thus universality in the bulk holds on a sequence close to , without having to assume that is a regular measure. Similar results are established for sequences of measures.