MathDB

We are given two point sets

A

and

B

which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in

B

is equal to

\dfrac{\pi}{m}

(

m \in \mathbb{N}

). We denote by

A^j

the set obtained by a counterclockwise rotation of

A

about the center of the unit circle for

\dfrac{j\pi}{m}

(

j=1,2,3,\dots

). Show that there exists a natural number

k

such that

l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)

.(Here

l(X)

denotes the sum of lengths of all disjoint arcs in the point set

X

)

Problem Statement