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 3 of 6: Choose the rotation to keep all sums distinct

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a+e^{i\theta}b\ne c+e^{i\theta}d\quad((a,b)\ne(c,d))

Detailed analysis

There are finitely many pairs of distinct pairs $(a,b),(c,d)$ . Each equality $a+e^{i\theta}b=c+e^{i\theta}d$ excludes at most finitely many values of $\theta$ (unless it is the identical pair). Choose $\theta$ outside this finite set, so $U$ has exactly $|S||T|$ distinct points.