  In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian
matrices the probability that an interval of length $s$ contains no eigenvalues
is the Fredholm determinant of the sine kernel $\sin(x-y)\over\pi(x-y)$ over
this interval. A formal asymptotic expansion for the determinant as $s$ tends
to infinity was obtained by Dyson. In this paper we replace a single interval
of length $s$ by $sJ$ where $J$ is a union of $m$ intervals and present a proof
of the asymptotics up to second order. The logarithmic derivative with respect
to $s$ of the determinant equals a constant (expressible in terms of
hyperelliptic integrals) times $s$, plus a bounded oscillatory function of $s$
(zero of $m=1$, periodic if $m=2$, and in general expressible in terms of the
solution of a Jacobi inversion problem), plus $o(1)$. Also determined are the
asymptotics of the trace of the resolvent operator, which is the ratio in the
same model of the probability that the set contains exactly one eigenvalue to
the probability that it contains none. The proofs use ideas from orthogonal
polynomial theory.
