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 2 of 6: Form a rotated sum

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U=S\oplus_\theta T=\{a+e^{i\theta}b:a\in S,b\in T\}

Detailed analysis

For finite unit-distance sets $S,T$ , translate one of them first if necessary so the two sets are disjoint, and define $U=S\oplus_\theta T=\{a+e^{i\theta}b:a\in S,b\in T\}$ . Translation preserves all unit distances.