Sumsets with distinct summands and the Erdõs-Heilbronn conjecture on sums of residues

Let S be a set of integers or of residue classes modulo a prime p, with cardinality |S| = k, and let T be the set of all sums of two distinct elements of S. For the integer case, it is shown that if |T| is less than approximately 2.5k then S is contained in an arithmetic progression with relatively small cardinality. For the residue class case a result of this type is derived provided that k > 60 and p > 50k. As an application, it is shown that |T| ≥ 2k-3 under these conditions. Earlier results of Freiman play an essential role in the proofs.