发表机构
University of Science and Technology of China (USTC); SIMIS, Fudan University; Shenzhen University（中国科学技术大学; 复旦大学上海数学中心; 深圳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究通过高斯整数商和艾森斯坦整数商对复数进行一致丢番图逼近的零一律，利用齐次动力学工具归结为收缩目标问题，通过不同论证得出临界轨迹邻域测度估计等新要素。
AI 中文摘要
我们建立了两个不同的零一律，用于通过高斯整数商和艾森斯坦整数商对复数进行一致丢番图逼近。利用齐次动力学工具，通过将其归结为\(\mathrm{SL}_2(\mathbb{C})\)某些齐次空间上的收缩目标问题来研究此问题。主要的新要素包括对相应临界轨迹的某类邻域的测度估计，以及用于控制短程混合贡献的新的不相交性陈述。由于高斯和艾森斯坦情形下临界轨迹的性质不同，这些测度估计是通过相当不同的论证得到的。
英文摘要
We establish two distinct zero-one laws for the uniform Diophantine approximation of complex numbers by quotients of Gaussian integers and by quotients of Eisenstein integers. Using tools from homogeneous dynamics, we study this problem by reducing it to a shrinking target problem on certain homogeneous spaces of $\mathrm{SL}_2(\mathbb{C})$. The main novel ingredients include measure estimates on a certain family of neighborhoods of the corresponding critical loci, as well as new disjointness statements to control the short-range mixing contribution. Due to the different nature of the critical loci in the Gaussian and Eisenstein cases, these measure estimates are obtained by rather different arguments.
CommentsMinor revision, 47 pages, comments welcome!
