On sumsets of multisets in \(\mathbb{Z}_p^m\)
type: Item
instance of: scholarly article
title: On sumsets of multisets in \(\mathbb{Z}_p^m\)
zbMATH Open document ID: 1296.11008
author: Kaisa Matomäki
published in: The Electronic Journal of Combinatorics
publication date: 2014-08-14T00:00:00Z
full work available at URL: v20i3p30
review text: 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)].
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 6330302
MaRDI profile type: Publication
cites work: Additive bases of vector spaces over prime fields
cites work: Group connectivity of graphs --- a nonhomogeneous analogue of nowhere-zero flow properties
cites work: Lower bounds for sumsets of multisets in Z_p^2
cites work: Addition theorems in elementary Abelian groups, I
cites work: Q3653943
cites work: Q4111952
description: scientific article; zbMATH DE number 6330302