  Let $\mathcal{A}$ denote a finite set of arithmetic progressions of positive
integers and let $s \geq 2$ be an integer. If the cardinality of $\mathcal{A}$
is at least 2 and $U$ is the union formed by taking certain arithmetic
progressions of length $s$ from each element of $\mathcal{A}$, we calculate the
asymptotic density of the set of all prime numbers $p$ such that $U$ is a set
of quadratic residues (respectively, quadratic non-residues) of $p$.
