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Let $\omega$ be a weight function defined on a locally compact group $G$,$1\le p<+\infty$, $S\subset G$ and let us assume that for any $s\in S$, theleft translation operator $T_s$ is continuous from the weighted $L^p$-space$L^p(G,\omega)$ into itself. For a given set $\Gamma\subset\mathbb{C}$, avector $f\in L^p(G,\omega)$ is said to be $(\Gamma,S)$-dense if the set $\{\lambda T_sf:\, \lambda\in \Gamma, \,s\in S\}$ is dense in $L^p(G,\omega)$. Inthis paper, we characterize the existence of $(\Gamma,S)$-dense vectors in$L^p(G,\omega)$ in terms of the weight and the set $\Gamma$.

$\Gamma$-supercyclicity of families of translates in weighted $L^p$-spaces on locally compact groups