An inverse theorem for the restricted set addition in Abelian groups

Let A be a set of k 5 elements of an Abelian group G in which the order of the smallest nonzero subgroup is larger than 2k−3. Then the number of different elements of G that can be written in the form a+a′, where a,a′ A, a≠a′, is at least 2k−3, as it has been shown in [Gy. Károlyi, The Erdős–Heilbronn problem in Abelian groups, Israel J. Math. 139 (2004) 349–359]. Here we prove that the bound is attained if and only if the elements of A form an arithmetic progression in G, thus completing the solution of a problem of Erdős and Heilbronn. The proof is based on the so-called ‘Combinatorial Nullstellensatz.’