Additive triples in groups of odd prime order

Let p be an odd prime. For nontrivial proper subsets A, B of Zₚ of cardinality s, t, respectively, we count the number r (A, B, B) of additive triples, namely elements of the form (a, b, a+b) in A B B. For given s, t, what is the spectrum of possible values for r (A, B, B)? In the special case A=B, the additive triple is called a Schur triple. Various authors have given bounds on the number r (A, A, A) of Schur triples, and shown that the lower and upper bound can each be attained by a set A that is an interval of s consecutive elements of Zₚ. However, there are values of p, s for which not every value between the lower and upper bounds is attainable. We consider here the general case where A, B can be distinct. We use Pollard's generalization of the Cauchy-Davenport Theorem to derive bounds on the number r (A, B, B) of additive triples. In contrast to the case A=B, we show that every value of r (A, B, B) from the lower bound to the upper bound is attainable: each such value can be attained when B is an interval of t consecutive elements of Zₚ.