Weighted Davenport’s constant and the weighted EGZ Theorem

Let G ∗ , G be finite abelian groups with nontrivial homomorphism group Hom ( G ∗ , G ) . Let Ψ be a non-empty subset of Hom ( G ∗ , G ) . Let D Ψ ( G ) denote the minimal integer, such that any sequence over G ∗ of length D Ψ ( G ) must contain a nontrivial subsequence s 1 , … , s r , such that ∑ i = 1 r ψ i ( s i ) = 0 for some ψ i ∈ Ψ . Let E Ψ ( G ) denote the minimal integer such that any sequence over G ∗ of length E Ψ ( G ) must contain a nontrivial subsequence of length | G | , s 1 , … , s | G | , such that ∑ i = 1 | G | ψ i ( s i ) = 0 for some ψ i ∈ Ψ . In this paper, we show that E Ψ ( G ) = | G | + D Ψ ( G ) − 1 .