Large Intersection Property for Limsup Sets in Metric Space

Abstract

We show that limsup sets generated by a sequence of open sets in compact Ahlfors s-regular $(0<s<\infty )$ space $(X,{\mathscr{B}},\mu ,\rho )$ belong to the classes of sets with large intersections with index λ, denoted by $\mathcal {G}^{\lambda }(X)$ , 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 γ2 and δ belong to $\mathcal {G}^{s-\delta -\gamma _{2}}(X)$ almost surely, and random covering sets with exponentially mixing property belong to $\mathcal {G}^{s_{0}}(X)$ almost surely, where s0 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.