A Weighted Erdős-Ginzburg-Ziv Theorem

Abstract

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct so that they can be considered as sets. If S is a sequence of m+n−1 elements from a finite abelian group G of order m and exponent k, and if $$W = {\left\{ {w_{i} } \right\}}^{n}_{{i = 1}}$$ is a sequence of integers whose sum is zero modulo k, then there exists a rearranged subsequence $${\left\{ {b_{i} } \right\}}^{n}_{{i = 1}}$$ of S such that $${\sum\nolimits_{i = 1}^n {w_{i} b_{i} = 0} }$$. This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when m = n and wi = 1 for all i, and confirms a conjecture of Y. Caro. Furthermore, we in part verify a related conjecture of Y. Hamidoune, by showing that if S has an n-set partition A=A1, . . .,An such that |wiAi| = |Ai| for all i, then there exists a nontrivial subgroup H of G and an n-set partition A′ =A′1, . . .,A′n of S such that $$H \subseteq {\sum\nolimits_{i = 1}^n {w_{i} {A}\ifmmode{'}\else$'$\fi_{i} } }$$ and $${\left| {w_{i} {A}\ifmmode{'}\else$'$\fi_{i} } \right|} = {\left| {{A}\ifmmode{'}\else$'$\fi_{i} } \right|}$$ for all i, where wiAi={wiai |ai∈Ai}.