Quadratic residues and non-residues in arithmetic progression

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