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Does there exist expression?

Posted: Sat Jul 22, 2017 9:33 am
For which positive integers $n$ does there exist expression

$$\mathbb{R}^2 = \bigcup_{m=1}^{\infty} A_m$$
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where each $A_m$ is a disk of radius $1$ such that each point $x \in \mathbb{R}^2$ belongs to either the boundary of some $A_m$ or to precisely $n$ interiors of the sets $A_1$ , $A_2$ , $\dots$ ?