Suppose a $\mathcal{H}^{1}$ measurable set $A\subset \mathbb{R}^{n}$ has positive Hausdorff density $\Theta^{1}(\mathcal{H}^{1},A,x)=c>0$ in a point $x\in A$.

If we have a decomposition $A=B\cup C$ with disjoint $B$ and $C$, can it happen that $\Theta^{1}(\mathcal{H}^{1},B,x)$ and $\Theta^{1}(\mathcal{H}^{1},C,x)$ do not exists? Can it even happen that the lower densities of these sets are both zero? Are there conditions on $A$ to prevent this?

More specifically: Let $C_{s,\epsilon}(x)$ denote the double cone in direction $s\in\mathbb{S}^{n-1}$ with opening angle $\epsilon$ at the point $x$.

Is it possible that $\Theta^{1}(\mathcal{H}^{1},A\cap C_{s,\epsilon}(x),x)$ does not exists for all $s\in \mathbb{S}^{n-1}$ and all $\epsilon\in (0,\epsilon_{s})$? ($\epsilon_{s}$ is some maximal angle depending on the direction $s$) Is it possible that the lower Hausdorff density vanishes for all these double cones?