  In this paper we start to investigate a new body of questions in additive
combinatorics. The fundamental Cauchy--Davenport theorem gives a lower bound on
the size of a sumset A+B for subsets of the cyclic group Zp of order p (p
prime), and this is just one example of a large family of results. Our aim in
this paper is to investigate what happens if we restrict the number of elements
of one set that we may use to form the sums. Here is the question we set out to
answer: given two subsets, A and B, does B have a subset C of bounded size such
that A+C is large, perhaps even comparable to the size of A+B? In particular,
can we get close to the lower bound of the Cauchy--Davenport theorem? Our main
results show that, rather surprisingly, in many circumstances it is possible to
obtain not merely an asymptotic version of the usual sumset bound, but even the
exact bound itself.
