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.