Does there exist an expression?

For which positive integers n does there exist expression

    \[\mathbb{R}^2 = \bigcup_{m=1}^{\infty} A_m\]

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 ?

(Euler Competition , 2017)

Solution [official]

For none n. Assume , on the contrary , that such expression exists for some value of n. Let us first note that the set of disks is countable hence the Lebesgue measure of the union of all the circles is 0.

Further, consider any disk A_i and let us show that the number of the other disks it intersects is finite. Indeed, each disk that intersects A_k would have to be a subset of the disk or radius 3 co-centric with A_k. All such disks together would cover an area of at most 9 \pi so that each point is covered by at most n disks. Hence the total area of disks that intersect A_k is at most 9 \pi n and therefore there are not more than 9n of them.

Now, let us introduce the function f:\mathbb{R}^2 \rightarrow \mathbb{N} whose value f(P) is the number of disks A_i that P belongs to. Note that

  • f(P)=n for almost all P \in \mathbb{R}^2
  • f(P) is constant on every connected component of \mathbb{R}^2 \setminus \bigcup \limits_{k=1}^{\infty} \partial A_k.

What remains to be shown is that this is impossible. Since the disk A_i intersects only with finitely many other disks , let us consider the circular arc between two such intersection points such that the arc XY in the figure above. But then obviously the value f(P) for points on one side of the arc and f(P) for points on the other side of the arc differ by 1. For the collections of disks that cover the two regions is different – A_i is only on one side of the circular arc.

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