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A weighted generalization of two theorems of Gao

Abstract:

Let G be a finite abelian group and let A⊆ℤ be nonempty. Let D A(G) denote the minimal integer such that any sequence over G of length D A(G) must contain a nontrivial subsequence s 1...s r such that Σ i=1r w is ifor some w i∈A. Let E A(G) denote the minimal integer such that any sequence over G of length E A(G) must contain a subsequence of length {pipe}G{pipe}, s 1...s {pipe}G{pipe}, such that Σ {pipe}G{pipe}i=1 w is i for some w i∈A. In this paper, we show that E A(G)={pipe}G{pipe}+D A(G)-1, confirming a conjecture of Thangadurai and the expectations of Adhikari et al. The case A={1} is an older result of Gao, and our result extends much partial work done by Adhikari, Rath, Chen, David, Urroz, Xia, Yuan, Zeng, and Thangadurai. Moreover, under a suitable multiplicity restriction, we show that not only can zero be represented in this manner, but an entire nontrivial subgroup, and if this subgroup is not the full group G, we obtain structural information for the sequence generalizing another non-weighted result of Gao. Our full theorem is valid for more general n-sums with n≥{pipe}G{pipe}, in addition to the case n={pipe}G{pipe}. © 2012 Springer Science+Business Media, LLC.