Large intersection property for limsup sets in metric space

TL;DR

This paper establishes conditions under which limsup sets in metric spaces have large intersection properties, providing lower bounds on their Hausdorff dimensions and applying these results to random fractals and covering sets.

Contribution

It introduces new criteria for limsup sets to belong to classes with large intersection properties in Ahlfors regular spaces, extending to random fractals and rectangle-generated sets.

Findings

Limsup sets have large intersection properties under certain conditions.

Lower bounds on Hausdorff dimension of limsup sets are established.

Random fractals and covering sets almost surely belong to classes with large intersection properties.

Abstract

We show that limsup sets generated by a sequence of open sets in compact Ahlfors $s$-regular space $(X,\mathscr{B},\mu,\rho)$ belong to the classes of sets with large intersections with index $\lambda$, 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 $\gamma_2$ and $\delta$ 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 $s_0$ 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.