论文标题
Hausdorff维度估计适用于Lagrange和Markov Spectra，Zaremba理论以及Fuchsian群体的极限集
Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups
论文作者
论文摘要
在本说明中，我们将描述一种简单且实用的方法，以在某些一维Markov迭代函数方案的限制设置的Hausdorff尺寸上获得严格的界限。总体问题引起了人们的关注，但我们特别关注Hausdorff维度在解决其他数学红色领域的猜想和问题中的价值的作用。作为我们的第一个应用，我们确认并经常加强对Lagrange和Markov Spectra在Diophantine分析中的差异的猜想，这些猜想出现在Matheus和Moreira Arxiv的工作中：1803.01230。作为第二次应用，我们（重新）验证并改善了与Zaremba猜想相关的估计数，用于数字理论，用于Bourgain-Kontorovich Arxiv的工作：1107.3776V2，Huang Arxiv：1310.3772V4和Kan Arxiv：1310.3772V4和Kan Arxiv：1604.044884。作为第三次几何应用，我们严格地将拉普拉斯（Laplacian）频谱的底部绑定为无限面积表面，如麦克穆伦（McMullen）所研究的示例所示。在估计极限集维度的所有方法中，都存在有关算法效率，所需的计算工作效率和界限严格的问题。我们使用的方法具有简单有效的优点，我们以直接实施的方式以第3节的方式呈现。
In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas red of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira arXiv:1803.01230. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain-Kontorovich arXiv:1107.3776v2, Huang arXiv:1310.3772v4 and Kan arXiv:1604.04884. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in section 3 in a way that is straightforward to implement.
