A dual approach to triangle sequences: a multidimensional continued fraction algorithm.

*(English)*Zbl 1102.11034Over the last 150 years, mathematicians have been looking for the “correct” notion of multidimensional continued fraction. The usual continued fraction of a real number is periodic if and only if it is a real quadratic irrationality. The dream has been to connect the periodicity of an \(n\)-dimensional continued fraction expansion with the property of dealing with real algebraic irrationalities of degree \(n\). Many authors have produced periodic \(n\)-dimensional algorithms which imply algebraicity, but no one so far has proved the converse, namely that the property of dealing with algebraic irrationalities of degree \(n\) will imply periodicity.

In the paper under review, the eight authors consider a type of multidimensional continued fraction algorithm by providing some geometric interpretation of the so-called triangle sequence and give some criterion for when a triangle sequence describes a uniquely defined pair of numbers. Explicit examples of both uniqueness and non uniqueness can then be given. The ergodic and the dynamical properties of the triangle sequence are also studied. More precisely, according to the authors, “the triangle sequence is topologically strongly mixing”.

The paper under review is highly technical and builds upon a previous paper of T. Garrity [J. Number Theory 88, No. 1, 86–103 (2001; Zbl 1015.11031)].

In the paper under review, the eight authors consider a type of multidimensional continued fraction algorithm by providing some geometric interpretation of the so-called triangle sequence and give some criterion for when a triangle sequence describes a uniquely defined pair of numbers. Explicit examples of both uniqueness and non uniqueness can then be given. The ergodic and the dynamical properties of the triangle sequence are also studied. More precisely, according to the authors, “the triangle sequence is topologically strongly mixing”.

The paper under review is highly technical and builds upon a previous paper of T. Garrity [J. Number Theory 88, No. 1, 86–103 (2001; Zbl 1015.11031)].

Reviewer: Claude Levesque (QuĂ©bec)