On sumsets of multisets in \(\mathbb{Z}_p^m\)
The electronic journal of combinatorics, Tome 20 (2013) no. 3
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.
@article{10_37236_3269,
author = {Kaisa Matom\"aki},
title = {On sumsets of multisets in {\(\mathbb{Z}_p^m\)}},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {3},
doi = {10.37236/3269},
zbl = {1296.11008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3269/}
}
Kaisa Matomäki. On sumsets of multisets in \(\mathbb{Z}_p^m\). The electronic journal of combinatorics, Tome 20 (2013) no. 3. doi: 10.37236/3269
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