Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Question

Background

Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by $$ x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}. $$ Let $$ \frac{p_n}{q_n} = [a_1, a_2, \dots, a_n] $$ be the $n$th convergent, and let $T^n x = [a_{n+1}, a_{n+2}, a_{n+3},\dots]$.

A well-known formula gives $$ x- \frac{p_n}{q_n} = \frac{(-1)^n \cdot x \cdot Tx \cdot T^2 x \cdot \dots \cdot T^{n}x}{q_n} = \frac{(-1)^n }{q_n(q_{n+1}+q_nT^{n+1} x)}. $$ This formula holds (with very minor changes, if any) for most kinds of one-dimensional continued fractions, including the various real variants, such as even continued fractions, odd continued fractions, Nakada's $\alpha$-continued fractions, etc., and - I believe - for Hurwitz's complex continued fractions as well.

The question

On page 132 of his book on Multidimensional Continued Fractions, Schweiger gives a generalization of the formula as a corollary to Perron's Identity; however, this formula is comparatively much more complicated.

Are there specific multi-dimensional continued fraction algorithms for which this formula simplifies considerably to something closer to what is seen for one-dimensional continued fractions? Do there exist other analogs of this formula for multi-dimensional continued fractions?

Motivation

Anton Lukyanenko and I worked on a paper studying continued fractions on the Heisenberg group (a two-dimensional complex system), and were surprised to see a simple analog of the formula in our work (Theorem 3.23 on page 22). We believe this is unique among multidimensional continued fractions, but we were not sure if there existed other simple analogs that were just not as well known.

Answer 1

One of natural generalizations of continued fractions (CF) is 3D Voronoi-Minkowski CF. For Voronoi construction see Delone & Fadeev's The theory of irrationalities of the third degree for Minkowski see Hancock, H. Development of the Minkowski geometry of numbers. Vol. 1 (ch. X). This object is nice from different points of view. But it is essentially 2D object.

We can consider classical CF as a chain or as broken line, while 3D CF is a plane graph. We can walk along this graph in different directions. The answer depends on your problem. You can try to find all minima of a linear form $x+\alpha y+\beta z$ and you'll go in one direction, you can try to minimize simultaneously two forms $x+\alpha y$ and $x+\beta z$, and then you must go in another direction.

There is no universal formula because time is 2-dimesional and there is no universal direction. But one good news. We have universal invariant measure $$dx_2\,dx_3\,dy_1\,dy_3\,dz_1\,dz_2\begin{vmatrix} 1 & x_2 & x_3\\ y_1 & 1 & y_3\\ z_1&z_2&1\\ \end{vmatrix}^{-3} $$ which is an analog of classical Gauss measure $$dx_2\,dy_1\begin{vmatrix} 1 & x_2 \\ y_1 & 1 \\ \end{vmatrix}^{-2}. $$ It arises in different contexts.

(See for example Kontsevich, M. L. & Suhov, Y. M. Statistics of Klein polyhedra and multidimensional continued fractions Pseudoperiodic topology, Amer. Math. Soc., 1999, 197, 9-27,

Karpenkov, O. On invariant Mobius measure and Gauss-Kuzmin face distribution ArXiv Mathematics e-prints, 2006

A. A. Illarionov The average number of relative minima of three-dimensional integer lattices of a given determinant. Izvestiya: Mathematics, 2012, 76:3, 535–562.)