Bulk universality holds in measure for compactly supported measures

Let μ be a measure with compact support, with orthonormal polynomials {p _n } and associated reproducing kernels {K _n }. We show that bulk universality holds in measure in {ξ: μ′(ξ) > 0}. More precisely, given ɛ, r > 0, the linear Lebesgue measure of the set {ξ: μ′(ξ) > 0} and for which

$mathop {sup }limits_{left| u right|,left| v right| leqslant r} left| {frac{{K_n (xi + u/tilde K_n (xi ,xi ),xi + v/tilde K_n (xi ,xi ))}} {{K_n (xi ,xi )}}} right. - left. {frac{{sin pi (u - v)}} {{pi (u - v)}}} right| geqslant varepsilon$mathop {sup }limits_{left| u right|,left| v right| leqslant r} left| {frac{{K_n (xi + u/tilde K_n (xi ,xi ),xi + v/tilde K_n (xi ,xi ))}} {{K_n (xi ,xi )}}} right. - left. {frac{{sin pi (u - v)}} {{pi (u - v)}}} right| geqslant varepsilon
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