Zero sums in restricted sequences

A sequence $\text{bfx}=(x_1,\dots ,x_m)$ of elements of ${\mathbb{Z}}_n$ is called an $A\mathit{\text{-weighted}}\mathit{\text{Davenport}}\mathit{\text{Z-sequence}}$ if there exists $\text{bfa}:=(a_1,\dots ,a_m)\in (A\cup \{0\}{)}^m\setminus {\text{bfzero}}_m$ such that $\sum_i a_ix_i=0$\mathrm{.}Here$\bfzero_m=(0,\ldots,0)\in\Z_n^m$\mathrm{.}Similarly,thesequence$\bfxis called an \textit{A-weighted Erd\H{o}s Z-sequence} if there exists\bfa:=(a_1,\ldots,a_m)\in (A\cup{0})^m\setminus{\bfzero_m}$with$|Supp(\bfa)|=n$,suchthat$\sum_i a_ix_i=0$,where$Supp(\bfa):={i: a_i\ne 0}$\mathrm{.}A$\Z_n$-sequence$\bfx$iscalled$k$-restrictedifnoelementof$\Z_n$appearsmorethan$k$timesin$\bfx$. In this paper, we study the problem of determining the least value of $m$ for which a $k$-restricted ${\mathbb{Z}}_n$-sequence of length $m$ is an $A$-weighted Davenport Z-sequence (resp. an$A$-weighted Erd\H{o}s Z-sequence). We also consider the same problem for random ${\mathbb{Z}}_n$ sequences, for certain very natural choices for the set $A$.