Rational approximations, multidimensional continued fractions and lattice reduction
Number Theory
2023-03-15 v1
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.
Cite
@article{arxiv.2303.07777,
title = {Rational approximations, multidimensional continued fractions and lattice reduction},
author = {Valerie Berthé and Karma Dajani and Charlene Kalle and Ela Krawczyk and Hamide Kuru and Andrea Thevis},
journal= {arXiv preprint arXiv:2303.07777},
year = {2023}
}
Comments
30 pages, 4 figures