for the positive real case and in [@GS] for the one-dimensional real case. We say the set of all such points in $A$ is the [*set of uniqueness*]{} and denote it by ${{\mathcal{U}}}_{-{\lambda}, \mu}$, ${{\mathcal{U}}}_{\nu}$ and ${{\mathcal{U}}}_{\kappa}$ for the mixed real case the Jordan block case, and the complex case respectively.

The purpose of this section is to provide a proof of Theorem \[thm:uniq\] by considering all three cases.

The main outline of all three of these proofs are the same:

-   find the vertices for the convex hull of $A$;

-   show that these vertices have unique addresses;

-   using these vertices, in combination with Lemma \[thm:unique tool\] below, construct a set of points with unique addresses that have positive Hausdorff dimension.

\[thm:unique tool\] Denote $\overline m=p, \overline p=m$ and assume that $u = a_1 a_2 \dots a_\ell$, $v = b_1 b_2 \dots b_k$ and $w = c_1 c_2 \dots c_n$ satisfy

-   $\pi [a_i a_{i+1} \dots a_\ell b_1 b_2 \dots b_k a_1 a_2 \dots a_\ell] \cap \pi [\overline{a_i}] = {\varnothing}$;

-   $\pi [b_j b_{j+1} \dots b_k a_1 a_2 \dots a_\ell] \cap \pi [\overline{b_j}] = {\varnothing}$;

-   $\pi [a_i a_{i+1} \dots a_\ell c_1 c_2 \dots c_n a_1 a_2 \dots 