Koji Momihara, Kumamoto

University(joint work with Masashi

Shinohara)

11-08-2015

　　　　

Distance sets on circles

Distance sets on spheres

It is well-known that

Def.

A regular polygon attains this bound as t=1.

A k-distance set on a circle lies

on a regular polygon if k is small enough.

Prob.

Any DS with is .

Distance sets on regular polygons

Regular Polygon

The number of distances

=The number of angles

=The number of length of arcs

≒(The number of differences as elements of )/2

Main Thm (M-Shinohara)

Thm.

Example

The bound is sharp.

Thm.

This bound is sharp!

How to get main Thm

2. Distance sets on

1. Partition of the unit circle The number of

distances=The number of length of arcs

∃a line through the origin partitioning X into

two parts of equal size.

How to get main Thm

2. Distance sets on

1. Partition of the unit circle

3. Fusion of two distance sets on

4. An application of Kneser’s addition Thm

(When does it lie on regular polygons?)

The number of distances

=The number of length of arcs

∃a line through the origin partitioning X into

two parts of equal size.

Prop.

4. An application of Kneser’s Thm

Thm (Kneser, 1953).

Cor.

4. An application of Kneser’s Thm

Cor.

Assume that .

・

Main Thm (M-Shinohara)

Thm.

Result.

Result.

Thanks for your attention!