Note on the Inverse Problems Associated with Subsequence Sums

Abstract

Let $G=C_n\oplus C_n$ with $n\ge 3$ and $S$ be a sequence with elements of $G$. Let $\mathrm{\Sigma }(S)\subseteq G$ denote the set of group elements which can be expressed as a sum of a nonempty subsequence of $S$. In this note, we show that if $S$ contains $2n–3$ elements of $G$, then either $0\in \mathrm{\Sigma }(S)$ or $|\mathrm{\Sigma }(S)|\ge n^2–n–1$. Moreover, we determine the structures of the sequence $S$ over $G$ with length $|S|=2n–3$ such that $0\notin \mathrm{\Sigma }(S)$ and $|\mathrm{\Sigma }(S)|=n^2–n–1$.