Generalised Gauss--Kuzmin Distribution for Klein sails in $\mathbb R^3$

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.