Abstract The elementary three distance theorem (Steinhaus conjecture) is specified so that the two or at most three distances are concretely represented by the marginal gaps and the gap that covers where α is the fractional part of a given irrational number. It is interesting to see that the orderly structure of the fractional parts of the first n multiples of an irrational number is completely represented by the marginal gaps. We better understand how the smaller marginal gap reduces the larger marginal gap and toggles as n increases. This aspect of the three distance theorem is tantamount to the rational approximation of an irrational number by the convergents of its continued fraction expansion.

中文翻译：

---

指定的三距离定理

摘要 基本的三距离定理（Steinhaus 猜想）被指定为两个或至多三个距离具体表示为边际间隙和覆盖其中 α 是给定无理数的小数部分的间隙。有趣的是，一个无理数的前 n 倍数的小数部分的有序结构完全由边际间隙表示。我们更好地理解较小的边际间隙如何减小较大的边际间隙并随着 n 的增加而切换。三距离定理的这一方面相当于无理数通过其连分数展开的收敛性的有理逼近。