On weighted zero-sum sequences
Abstract
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 cii=1m with terms from G has a length n=exp(G) subsequence cijj=1n for which there are a1,...,an in A such that sumj=1naicij=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) is at most 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,-1, we determine the asymptotic behavior of s1,-1(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s1,-1(G)=exp(G)+log2|G|+O(log2log2|G|) as exp(G) tends to the infinity. Combined with a lower bound of exp(G)+sumi=1r2 ni, where G=n1... nr with 1<n1|... |nr, this determines s1,-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 s1,-1(G) are also computed.