Weighted Davenport’s constant and the weighted EGZ Theorem

Let G∗,GG∗,G be finite abelian groups with nontrivial homomorphism group Hom(G∗,G). Let ΨΨ be a non-empty subset of Hom(G∗,G). Let DΨ(G)DΨ(G) denote the minimal integer, such that any sequence over G∗G∗ of length DΨ(G)DΨ(G) must contain a nontrivial subsequence s1,…,srs1,…,sr, such that ∑i=1rψi(si)=0 for some ψi∈Ψψi∈Ψ. Let EΨ(G)EΨ(G) denote the minimal integer such that any sequence over G∗G∗ of length EΨ(G)EΨ(G) must contain a nontrivial subsequence of length |G|,s1,…,s|G||G|,s1,…,s|G|, such that ∑i=1|G|ψi(si)=0 for some ψi∈Ψψi∈Ψ. In this paper, we show that EΨ(G)=|G|+DΨ(G)−1.EΨ(G)=|G|+DΨ(G)−1.