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By Sukumar Das Adhikari, David J. Grynkiewicz and Zhi-wei Sun


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} m i=1 with terms from G has a length n = exp(G) subsequence {cij} n j=1 for which there are a1,...,an ∈ A such that ∑n j=1aici j = 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)+log 2 |G|+O(log 2 log 2 |G|) as exp(G) → +∞. Combined with a lower bound of exp(G) + ∑ r i=1 ⌊log 2 ni⌋, 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

Year: 2010
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