On restricted sumsets over a field

We consider restricted sumsets over field F. LetC={a1+⋯+an:a1∈A1,…,an∈An,ai−aj∉Sijifi≠j}, where Sij(1⩽i≠j⩽n) are finite subsets of F with cardinality m, and A1,…,An are finite nonempty subsets of F with |A1|=⋯=|An|=k. Let p(F) be the additive order of the identity of F. It is proved that |C|⩾min⁡{p(F),n(k−1)−mn(n−1)+1} if p(F)>mn. This conclusion refines the result of Hou and Sun [11].