A Weighted Erdős-Ginzburg-Ziv Theorem

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 w _i = 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, . . .,A _n such that |w _i A _i | = |A _i | 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 w _i A _i ={w _i a _i |a _i ∈A _i }.