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 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"[email protected]?A such that @?"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)= +~. Combined with a lower bound of exp(G)[email protected]?"i"="1^[email protected]?log"2n"[email protected]?, where [email protected]?Z"n"""[email protected][email protected]?Z"n"""r with 1

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