On the intersection of dynamical covering sets with fractals

Let \((X,\mathscr {B}, \mu ,T,d)\) be a measure-preserving dynamical system with exponentially mixing property, and let \(\mu \) an Ahlfors s-regular probability measure. The covering problem concerns the set E(x) of points which are covered by orbits \(x\in X\) infinitely many times. We prove that Hausdorff dimension intersection any regular fractal G \(\dim _\mathrm{H}G>s-\alpha equals _\mathrm{H}G+\alpha -s\), where \(\alpha =\dim _\mathrm{H}E(x)\) \)–a.e. Moreover, we obtain packing \(E(x)\cap G\) estimate for _\mathrm{H}(E(x)\cap G)\) analytic G.