Lagrange spectrum of a circle over the Eisensteinian field

Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

Let $\mathscr{L}(S^1)$ be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle $S^1$ by its rational points. We give a complete description of the structure of $\mathscr{L}(S^1)$ below its smallest accumulation point. To this end, we use digit expansions of points on $S^1$, which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of $\mathscr{L}(S^1)$ is 2. Also we characterize the points on $S^1$ whose Lagrange numbers are less than 2 in terms of Romik's di…

圆与球面上的内蕴丢番图逼近

Intrinsic Diophantine approximation on circles and spheres

本文提出统一框架，将六个圆和球面上的内蕴丢番图逼近的拉格朗日谱与实数域和复数域的谱关联，从而完整确定各谱的最小聚点及初始离散部分。

Hecke群H4上的Markoff谱和Lagrange谱

The Markoff and Lagrange spectra on the Hecke group H4

本文研究Hecke群H4上的Markoff谱和Lagrange谱，发现其在第一个积累点后具有正Hausdorff维数，并确定了谱中的间隙及Hall射的界。