Rational approximations, multidimensional continued fractions and lattice reduction

TL;DR

This paper surveys multidimensional continued fraction algorithms and lattice reduction, analyzing their convergence and approximation quality, and introduces a Jacobi--Perron based algorithm with potential ergodic properties.

Contribution

It provides a comprehensive comparison of dynamical properties of multidimensional continued fractions and lattice reduction, and proposes a new approach to establish ergodic measures for a Jacobi--Perron based algorithm.

Findings

Comparison of convergence properties of different algorithms

Analysis of the approximation quality of these algorithms

Proposal of a method to prove ergodic invariant measures

Abstract

We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction. We discuss their convergence properties and the quality of the rational approximation, and stress the interest for these algorithms to be obtained by iterating dynamical systems. We then focus on an algorithm based on the classical Jacobi--Perron algorithm involving the nearest integer part. We describe its Markov properties and we suggest a possible procedure for proving the existence of a finite ergodic invariant measure absolutely continuous with respect to Lebesgue measure.