论文标题
最差的合理数字
The worst approximable rational numbers
论文作者
论文摘要
我们使用双曲几何形状将所有理性数字与近似常数分类至少至少$ \ frac {1} {3} $。理性数量对应于模块化圆环中的大地学，两端都在尖端中，近似常数衡量它们在介于两者之间的距离远离尖口。与原始方法相比，几何观点消除了讨论持续分数表示的复杂符号动态的需求，并且它阐明了两种类型的最差的近似理性理性的区别：（1）有马尔可夫分数的平面森林的分母是Markov数字。它们对应于模块化的圆环中的简单大地学，两端都在尖端。 （2）对于每个马尔可夫级分，有两个无限的伴侣序列，它们对应于cusp中两端的非简单大地测量学序列，它们不会与一对脱节简单的大地测量学相交，一个在尖端中两端，一个都关闭。
We classify and enumerate all rational numbers with approximation constant at least $\frac{1}{3}$ using hyperbolic geometry. Rational numbers correspond to geodesics in the modular torus with both ends in the cusp, and the approximation constant measures how far they stay out of the cusp neighborhood in between. Compared to the original approach, the geometric point of view eliminates the need to discuss the intricate symbolic dynamics of continued fraction representations, and it clarifies the distinction between the two types of worst approximable rationals: (1) There is a plane forest of Markov fractions whose denominators are Markov numbers. They correspond to simple geodesics in the modular torus with both ends in the cusp. (2) For each Markov fraction, there are two infinite sequences of companions, which correspond to non-simple geodesics with both ends in the cusp that do not intersect a pair of disjoint simple geodesics, one with both ends in the cusp and one closed.
