The distribution of sequences in residue classes

We prove that any set of integers A subset of [1, x] with \A\ >> (log x)(r) lies in at least nu(A)(p) >> p(r/r+1) many residue classes modulo most primes p << (log x)(r+1). (Here r is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below x which are multiplicatively generated by the coprime integers a(1),...,a(r) (i.e. whose counting function is also c(log x)(r)) lie in at least p(r/r+1+epsilon(p)) residue classes, modulo most small primes p, where epsilon(p) --> 0, as p --> infinity.