1. Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles

2. 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.

3. 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

4. Main Thm (M-Shinohara) Thm. Exampl e

5. The bound is sharp. Thm. This bound is sharp!

6. 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.

7. 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.

8. 4. An application of Kneser’s Thm Thm (Kneser, 1953). Cor.

9. 4. An application of Kneser’s Thm Cor. Assume that. ・

10. Main Thm (M-Shinohara) Thm. Resul t. Thanks for your attention!