connected. Let $a, b \in A_{{\kappa}^2}$ and $\gamma$ the path connecting them. Consider ${\kappa}a, {\kappa}b \in {\kappa}A_{{\kappa}^2}$ and let $\gamma'$ be the path between them. As ${\kappa}\not \in {\mathbb{R}}$, we see that $\gamma$ and $\gamma'$ cannot be parallel lines. By observing that $\sum a_i \kappa^i = \sum a_{2i} \kappa^{2i} + {\kappa}\sum a_{2i +1} \kappa^{2i}$, we have $A_{\kappa}= A_{{\kappa}^2} + {\kappa}A_{{\kappa}^2}$ (the Minkowski sum). In particular, $A_{\kappa}$ will contain $\gamma + \gamma'$. By Theorem \[thm:Victor\] below, $\gamma+ \gamma'$ contains points in its interior, whence so does $A_{\kappa}$.

Hence if $|{\kappa}|\ge 2^{-1/4}$, then $A_{\kappa}$ has non-empty interior.

A great deal of information is known about the set $\mathcal{M}$ of all ${\kappa}$ for which $A_{\kappa}$ is connected – see [@Cal] and references therein.

The following proof is by V. Kleptsyn (via Mathoverflow [@MO]).

\[thm:Victor\] If $\gamma$ and $\gamma'$ are two paths in ${\mathbb{R}}^2$, not both parallel lines, then $\gamma + \gamma'$ has non-empty interior.

See Appendix.

Unique addresses and convex hulls {#sec:unique}
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Recall that a point $(x,y) \in A$ has a unique address (notation: $(x,y)\in\mathcal U$) if there is a unique sequence $(a_i)_0^\infty \in \{p, m\}^{\mathbb N}$ such that $(x,y) = \pi (a_1 a_2 a_3 \dots)$. These have been studied in [@HS] 