AI 中文总结
本文将经典高斯-库兹明分布推广至三维空间，证明一般克莱因帆局部面结构的等分布，解决Karpenkov公开问题，通过构造SL₃(ℝ)/SL₃(ℤ)对角流的有限截面测度推导极限面统计量。
AI 中文摘要
经典高斯-库兹明分布描述连分数数位的渐近分布，从几何角度可解释为二维空间中克莱因帆面的等分布。本文建立该现象的三维类似情形，证明三维空间中一般克莱因帆的局部面结构呈等分布，从而得到高斯-库兹明分布的高维推广。此外，本文解决了Karpenkov提出的若干公开问题。本文的方法基于齐性动力学，受Kontsevich与Suhov的工作启发，具体而言，本文在SL₃(ℝ)/SL₃(ℤ)上的对角流中构造一个截面，使得对该截面的访问可编码三维克莱因帆的几何结构。主要技术贡献在于证明相关截面测度是有限的，这使得能够推导极限面统计量。
英文摘要
The classical Gauss--Kuzmin distribution describes the asymptotic distribution of continued fraction digits. Geometrically, this may be interpreted as the equidistribution of faces of Klein sails in $\mathbb R^2$. In this paper, we establish a three-dimensional analogue of this phenomenon. We prove the equidistribution of local face structures in generic Klein sails in $\mathbb R^3$, thereby obtaining a higher-dimensional generalization of the Gauss--Kuzmin distribution. In addition, we resolve several open questions posed by Karpenkov~\cite{Ka17}. Our approach is based on homogeneous dynamics and is motivated from the work of Kontsevich and Suhov~\cite{KS99}. More precisely, we construct a cross-section for the diagonal flow on $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$, such that visits to the cross-section encode the geometry of three-dimensional sails. The principal technical contribution is the proof that the associated cross-sectional measure is finite, which enables us to derive the limiting face statistics.
Comments50 pages
