On weighted zero-sum sequences


Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer m, denoted by sA(G), such that any sequence { ci}i=1m with terms from G has a length n=exp(G) subsequence { cij}j=1n for which there are a1,..., an∈A such that Σj=1n aicij=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that sA(G)≤⌈D(G)/|A|⌉+exp(G)-1 if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for sA(G) in terms of |A| is essentially best possible. In the case A={±1}, we determine the asymptotic behavior of s{ ±1}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank,s{ ±1}(G)=exp(G)+ log2|G|+O( log2log2|G|)asexp(G)→+∞. Combined with a lower bound of exp(G)+Σi=1r⌊ log2ni⌋, where G≅Z n1&⋯ & Znr with 1< n1|⋯| nr, this determines s{ ±1}(G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems. Some additional more specific values and results related to s{ ±1}(G) are also computed. © 2011 Elsevier Inc. All rights reserved.

Publication Title

Advances in Applied Mathematics