Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on...
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![Page 1: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/1.jpg)
Koji Momihara, Kumamoto
University(joint work with Masashi
Shinohara)
11-08-2015
Distance sets on circles
![Page 2: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/2.jpg)
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 .
![Page 3: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/3.jpg)
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
![Page 4: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/4.jpg)
Main Thm (M-Shinohara)
Thm.
Example
![Page 5: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/5.jpg)
The bound is sharp.
Thm.
This bound is sharp!
![Page 6: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/6.jpg)
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.
![Page 7: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/7.jpg)
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.
![Page 8: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/8.jpg)
4. An application of Kneser’s Thm
Thm (Kneser, 1953).
Cor.
![Page 9: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/9.jpg)
4. An application of Kneser’s Thm
Cor.
Assume that .
・
![Page 10: Koji Momihara, Kumamoto University (joint work with Masashi Shinohara) 11-08-2015 Distance sets on circles.](https://reader035.fdocuments.us/reader035/viewer/2022062423/56649eea5503460f94bfb7b6/html5/thumbnails/10.jpg)
Main Thm (M-Shinohara)
Thm.
Result.
Result.
Thanks for your attention!