Long gaps in sieved sets

For each prime p, let Ip ⊂ Z/pZ denote a collection of residue classes modulo p such that the cardinalities [Ip] are bounded and about 1 on average.We show that for sufficiently large x, the sifted set {n ϵ Z: n (mod p) ϵ Ip for all p ≤ x} contains gaps of size at least x(log x)δ where δ > 0 depends only on the density of primes for which Ip ≠ ø. This improves on the "trivial"bound of » x. As a consequence, for any non-constant polynomial f: Z → Z with positive leading coefficient, the set {n ≤ X: f (n) compositeg contains an interval of consecutive integers of length ≥ (logX)(log logX)δ for sufficiently large X, where δ > 0 depends only on the degree of f.