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.
Categories: math.NT