On Weighted Zero - Sum Sequences Sukumar

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=1 with terms from G has a length n = exp(G) subsequence {cij}nj=1 for which there are a1, . . . , an ∈ A such that ∑n j=1 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) ≤ dD(G)/|A|e + 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(log2 log2 |G|) as exp(G)→ +∞. Combined with a lower bound of exp(G) + ∑r i=1blog2 nic, where G ∼= Zn1 ⊕ · · · ⊕ 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.