Set boundary preserved by an infinite union

Question

Suppose I have a subset $U\subset\mathbb R^2$ and a real number $r>1$ with the following properties:

1. $U$ is compact;
2. $U\subset rU$ (self-similarity);
3. $0\in U$;
4. 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).

Answer 1

No.

Consider $U=\{(x,y)\mid x^2+(y-1)^2\le 1\lor x^2+(y+1)^2\le 1\}$ (union of two touching closed disks) and $r=2$. Then $V$ is dense in $\mathbb R^2$.