Let G be a finite abelian group of order n and let A ⊆ Z be non-empty.

A prototype of zero-sum theorems, the well-known theorem of Erdős, Ginzburg and Ziv says that for any positive integer n, any sequence a1,a2,…,a2n-1 of 2n-1 integers has a subsequence of n elements whose sum is 0 modulo n. Appropriate generalizations of the question, especially that for (Z/pZ)d, generated a lot of research and still have challenging open questions. Here we propose a new generalization of the Erdős–Ginzburg–Ziv theorem and prove it in some basic cases

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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 {ci j } n j=1 for which there are a1, . . . , an ∈ A such that n j=1 aici j = 0. 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,Combined with a lower bound of exp(G) + r i=1 ⌊log 2 ni⌋, where G ∼ = Zn 1 ⊕ · · · ⊕ Zn r 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.

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