## 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.