On the packing dimension of image set

1. There exists compacet set $E\subset [0,1]$ with $\dim_H E<\dim_P E$ (in fact, for any $\beta\in(0,1)$, $\dim_HE=0, \dim_P E_{\beta}=\beta$ ), and under any Holder function of order $\alpha$ is strictly smaller than $\frac{1}{\alpha}\dim_P E$.

2. Let $E\subset R$ be compact. For any $\alpha<\dim_P E$, there exists a Moran set $E_{\alpha}\subset E$ with $\dim_P E_{\alpha}\ge \alpha.$

 

See Fractional Brownian motion and packing dimension by Talagrand and Xiao, 1996.