On the number of weighted zero-sum subsequences

Let G be a finite additive abelian group with exponent dkn,d,n>1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^kn, d,n>1,$$\end{document} and k a positive integer. For a sequence S over G and A={1,2,…,dkn-1}\{dkn/di:i∈[1,k]},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=\{1,2,\ldots ,d^kn-1\}\setminus \{d^kn/d^i:i\in [1,k]\}, $$\end{document} we investigate the lower bound of the number NA,0(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{A,0}(S)$$\end{document}, which denotes the number of A-weighted zero-sum subsequences of S. In particular, we prove that NA,0(S)≥2|S|-DA(G)+1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{A,0}(S)\ge 2^{|S|-D_A(G)+1},$$\end{document} where DA(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_A(G)$$\end{document} is the A-weighted Davenport Constant. We also characterize the structures of the extremal sequences for which equality holds for some groups.