Quadratic non-residues in short intervals

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes p in a dyadic interval [Q, 2Q] for which a given interval [u + 1, u + ψ(Q)] does not contain a quadratic non-residue modulo p. The bound is non-trivial for any function ψ(Q)→ ∞ as Q→ ∞. This is an analogue of the well-known estimates on the smallest quadratic nonresidue modulo p on average over primes p, which corresponds to the choice u = 0.