# Erdős-Ulam 問題

## 有理距離嘅大集

Erdős-Anning 定理話一個集如果啲距離全部都係整數嘅話，個集一係就有限一係就成一直線。[1]但係，如果放鬆到有理數距離嘅話，就可以有其他嘅無限集，例如，喺單位圓上邊，設${\displaystyle S}$ 係由

${\displaystyle (\cos \theta ,\sin \theta )}$

${\displaystyle \left|2\sin {\frac {\theta }{2}}\cos {\frac {\varphi }{2}}-2\sin {\frac {\varphi }{2}}\cos {\frac {\theta }{2}}\right|}$

## 參考資料

1. Anning, Norman H.; Erdős, Paul (1945), "Integral distances", Bulletin of the American Mathematical Society, 51 (8): 598–600, doi:10.1090/S0002-9904-1945-08407-9.
2. Klee, Victor; Wagon, Stan (1991), "Problem 10 Does the plane contain a dense rational set?", Old and New Unsolved Problems in Plane Geometry and Number Theory, Dolciani mathematical expositions,第11卷, Cambridge University Press, pp. 132–135, ISBN 978-0-88385-315-3.
3. Solymosi, József; de Zeeuw, Frank (2010), "On a question of Erdős and Ulam", Discrete and Computational Geometry, 43 (2): 393–401, arXiv:0806.3095, doi:10.1007/s00454-009-9179-x, MR 2579704, S2CID 15288690
4. Tao, Terence (2014-12-20), "The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture", What's new, 喺2016-12-05搵到
5. Shaffaf, Jafar (May 2018), "A solution of the Erdős–Ulam problem on rational distance sets assuming the Bombieri–Lang conjecture", Discrete & Computational Geometry, 60 (8): 283–293, arXiv:1501.00159, doi:10.1007/s00454-018-0003-3, S2CID 51907500
6. Pasten, Hector (2017), "Definability of Frobenius orbits and a result on rational distance sets", Monatshefte für Mathematik, 182 (1): 99–126, doi:10.1007/s00605-016-0973-2, MR 3592123, S2CID 7805117