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Hans Publishers, Advances in Applied Mathematics, 3(48), p. 506-527

DOI: 10.1016/j.aam.2011.11.007

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On weighted zero-sum sequences

Journal article published in 2010 by Sukumar Das Adhikari, David J. Grynkiewicz, Zhi-Wei Sun
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Red circle
Preprint: archiving forbidden
Red circle
Postprint: archiving forbidden
Red circle
Published version: archiving forbidden
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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 s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,.,a_n in A such that sum_{j=1}^na_ic_{i_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 s_A(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 s_A(G)in terms of |A| is essentially best possible. In the case A={1,-1}, we determine the asymptotic behavior of s_{{1,-1}}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s_{{1,-1}}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends to the infinity. Combined with a lower bound of $exp(G)+sum{i=1}{r}⌊\log_2 n_i⌋$, where $G=\Z_{n_1}⊕.⊕ \Z_{n_r}$ with 1