Inverse problems associated with subsequence sums in Cp ⊕ Cp II

Abstract

Let G be a finite abelian group and S a sequence with elements of G. Let Σk(S)⊂G denote the set of group elements which can be expressed as a sum of a nonempty subsequence of S with length k. In this paper, we study the set Σp2(S) of the sequence S over Cp⊕Cp, where p is a prime and |S|=p2+r with p−1≤r≤2p−3. We prove that either 0∈Σp2(S) or |Σp2(S)|≥p(r−p+3)−1; furthermore, we determine the structure of S if 0∉Σp2(S) and |Σp2(S)|=p(r−p+3)−1. These results confirm some cases of two conjectures on p2-sums of a sequence over Cp⊕Cp.