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For a sequence $A$ of given length $n$ contained in $\mathbb{Z}_p^2$ we study how many distinct subsums $A$ must have when $A$ is not "wasteful" by containing too many elements in same subgroup. Martin, Peilloux and Wong have made a conjecture for a sharp lower bound and established it when $n$ is not too large whereas Peng has previously established the conjecture for large $n$. In this note we build on these earlier works and add an elementary argument leading to the conjecture for every $n$.Martin, Peilloux and Wong also made a more general conjecture for sequences in $\mathbb{Z}_p^m$. Here we show that the special case $n = mp-1$ of this conjecture implies the whole conjecture and that the conjecture is equivalent to a strong version of the additive basis conjecture of Jaeger, Linial, Payan and Tarsi.

On Sumsets of Multisets in $\mathbb{Z}_p^m$