  Let S be an infinite set of non-empty, finite subsets of the nonnegative
integers. If p is an odd prime, let c(p) denote the cardinality of the set {T
{\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues
(respectively, non-residues) of p}. When S is constructed in various ways from
the set of all arithmetic progressions of nonnegative integers, we determine
the sharp asymptotic behavior of c(p) as p {\to} +{\infty}. Generalizations and
variations of this are also established, and some problems connected with these
results that are worthy of further study are discussed.
