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Large intersection property for limsup sets in metric space

We show that limsup sets generated by a sequence of open sets in compact Ahlfors $s$-regular space $(X,\mathscr{B},μ,ρ)$ belong to the classes of sets with large intersections with index $λ$, denoted by $\mathcal{G}^λ(X)$, under some conditions. In particular, this provides a lower bound on Hausdorff dimension of such sets. These results are applied to obtain that limsup random fractals with indices $γ_2$ and $δ$ belong to $\mathcal{G}^{s-δ-γ_2}(X)$ almost surely, and random covering sets with exponentially mixing property belong to $\mathcal{G}^{s_0}(X)$ almost surely, where $s_0$ equals to the corresponding Hausdorff dimension of covering sets almost surely. We also investigate the large intersection property of limsup sets generated by rectangles in metric space.