Suppose I have a subset $U\subset\mathbb R^2$ and a real number $r>1$ with the following properties:
- $U$ is compact;
- $U\subset rU$ (self-similarity);
- $0\in U$;
- there exists an open set $H\subset \mathbb R^2$ such that $0\in\partial H$ (boundary) and $H\cap U=\emptyset$.
Let $V:=\bigcup_{n\geq0} r^n U$.
Question: Whether exists $H_1$ open such that $0\in\partial H_1$ and $H_1\cap V=\emptyset$.
Of course, if the answer is "yes", I would like to see a way how to prove it (it needn't be a complete proof, just some crucial hint).