Fractal sets and dimensions

Question

Can we construct two sets $E$ and $F$ meeting the following criteria

1. $\dim_H(E) = \dim_H(F) = \dim_H(E ∩ F)$

2. $\dim_P(E), \dim_P(F)$, and $\dim_P(E ∩ F)$ are distinct?

Here $\dim_H$ denotes the Hausdorff dimension and $\dim_P$ denotes the packing dimension.

Answer 1

Suppose we construct sets $A,B,C$ with
$A \subseteq [0,1]$, $\dim_\mathrm{H}(A) = 1/5$ and $\dim_\mathrm{P}(A) = 4/5$,
$B \subseteq [2,3]$, $\dim_\mathrm{H}(B) = 1/5$ and $\dim_\mathrm{P}(B) = 2/5$,
$C \subseteq [4,5]$, $\dim_\mathrm{H}(C) = 1/5$ and $\dim_\mathrm{P}(C) = 3/5$.
Let $E = A \cup B, F = B \cup C$, so that $E \cap F = B$.

Then
$\dim_\mathrm{H}(E) = \dim_\mathrm{H}(F) = \dim_\mathrm{H}(E\cap F) = 1/5$,
and $\dim_\mathrm{P}(E)=4/5, \dim_\mathrm{P}(F)=3/5, \dim_\mathrm{P}(E\cap F) = 2/5$.