A two-dimensional continued fractions algorithm with Lagrange and Dirichlet properties

TL;DR

This paper introduces a new two-dimensional continued fractions algorithm that satisfies Lagrange and Dirichlet properties, supported by geometric insights and convergence proofs, advancing multidimensional number theory methods.

Contribution

It presents a novel 2D continued fractions algorithm with proven Lagrange and Dirichlet properties, based on geometric principles, filling gaps in multidimensional continued fractions research.

Findings

Proved a Lagrange Theorem for a specific 2D algorithm.

Established Dirichlet-type convergence properties.

Provided geometric interpretation and validation of the algorithm.

Abstract

A Lagrange Theorem in dimension 2 is proved, for a particular two-dimensional algorithm, with a very natural geometrical definition. Dirichlet-type properties for the convergence of the algorithm are also proved. These properties procced from a geometrical quality of the algorithm. Some refercences are given to the works of various authors, in the domain of multidimensional continued fractions algorithms.