Problem 5

Prove that for every natural number

m

, there exists a finite set

S

of points in a plane such that every point

A

S

has exactly

m

points in

S

at unit distance from

A

Step 6 of 6: Induct from one edge to every m

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S_1=\{0,1\},\qquad S_m=S_{m-1}\oplus_\theta S_1

Detailed analysis

The set $S_1=\{0,1\}$ is $1$ -regular. If $S_{m-1}$ is $(m-1)$ -regular, apply the rotated-sum construction with $T=S_1$ and choose a generic $\theta$ . The resulting finite set $S_m=S_{m-1}\oplus_\theta S_1$ is $m$ -regular. Induction proves the claim for every natural $m$ .