Restricted sumsets in a finite abelian group

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