Does anyone have an exact reference for the weak convergence (convergence in angle) of Brun's subtractive multi-dimensional continued fractions algorithm (in all dimensions)? I have been given Schweiger's book on multi-dimensional continued fractions as a reference. However, perhaps the area is a bit foreign to me so that I could not exactly find it in there.
Thank you very much!
In section 5 of Brun Expansions, Substitutions and Discrete Geometry a ``$d$-dimensional" Brun map is introduced:
$$ T: (a_1, \dots, a_n) \mapsto (\tfrac{a_1}{a_k}, \dots, \{\tfrac{1}{a_j} \},\dots,\tfrac{a_n}{a_k})$$
where $\{ x\} \equiv x \mod 1$. Maps of this kind usually arise as an accelerated version of the Euclidean algorithm.
We could imagine a map $R: x \mapsto x + a$ with initial value $0$.
Consider the first-return map on the interval $[0,R(0)=a]$. We need to find a number such that
$$ ma < a \mod 1$$
In fact, $m = \lfloor \tfrac{1}{a} \rfloor+1$. If we rescale the interval so that $[0,a]$ is now $[0,1]$. In that case
$$ 0 \mapsto \lfloor \tfrac{1}{a} \rfloor \mod \tfrac{1}{a} \equiv \{ \tfrac{1}{a}\}$$
So our rotation got rescaled to $R_1: x \mapsto x + \{ \tfrac{1}{a}\}$. This renormalization is acting on the set of rotations.
It's likely the Brun expansion arises from a "Euclidean algorithm" on vectors.
Here is another interesting question: Do simple, multi-dimensional generalizations of this continued fraction formula exist?
Google searching has found many interesting variants on the continued fraction algorithm:
The more I read the more ambiguous the term "multidimensional continued fraction" gets.
The original 1957 article by Brun himself article is clearly written. It contains full, clean proofs of weak convergence and of some linear independence results (at least for dimension 2).
But the article is only available in French, in the proceedings of the XIIIth Congress of Scandinavian Mathematicians (1957). Fortunately, there is a copy of these proceedings in IHP (Paris). Here is a scanned copy: http://jolivet.org/timo/docs/Brun_algo.pdf
