On sumsets of multisets in \(\mathbb{Z}_p^m\)
Summary: 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. \textit{G. Martin, A. Peilloux} and \textit{E. B. Wong} [Integers 13, Paper A72, 17 p. (2013; Zbl 1295.11011)] have made a conjecture for a sharp lower bound and established it when \(n\) is not too large whereas \textit{C. Peng} [J. Number Theory 27, 46--57, 58--62 (1987; Zbl 0624.10046)] 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 et al. [loc. cit.] 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 \textit{F. Jaeger} et al. [J. Comb. Theory, Ser. B 56, No. 2, 165--182 (1992; Zbl 0824.05043)].
scientific article; zbMATH DE number 6330302