Restricted sumsets in a finite abelian group

In this paper, we prove that if AA and BB are subsets of a finite abelian group GG with |A|+|B|=|G|+L(G)|A|+|B|=|G|+L(G), then |A+ˆB|≥|G|−2, where L(G)=|{g:g∈G,2g=0}|L(G)=|{g:g∈G,2g=0}| andA+ˆB={a+b:a∈A,b∈B,a≠b}. In addition, we give a complete description of the subsets AA and BB of GG such that |A|+|B|=|G|+L(G)|A|+|B|=|G|+L(G) and A+ˆB≠G. Our results generalize the corresponding theorems of Gallardo et al. in cyclic group Z/nZZ/nZ [L. Gallardo, G. Grekos, L. Habsieger, et al., Restricted addition in Z/nZZ/nZ and an application to the Erdös–Ginzburg–Ziv problem, J. London Math. Soc. 65 (2) (2002) 513–523].