Berthé, V. and Dajani, K. and Kalle, C. and Krawczyk, E. and Kuru, Hamide and Thevis, A. (2024) Rational approximations, multidimensional continued fractions, and lattice reduction. In: Abdellatif, Ramla and Karemaker, Valentijn and Smajlovic, Lejla, (eds.) Women in Numbers Europe IV: Research Directions in Number Theory. Association for Women in Mathematics Series, 32. Springer Cham, Switzerland, pp. 111-154. ISBN 978-3-031-52162-1 (Print) 978-3-031-52163-8 (Online)
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Official URL: https://dx.doi.org/10.1007/978-3-031-52163-8_5
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 generated rational approximations 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.
| Item Type: | Book Section / Chapter |
|---|---|
| Divisions: | Faculty of Engineering and Natural Sciences |
| Depositing User: | Hamide Kuru |
| Date Deposited: | 03 Sep 2024 12:42 |
| Last Modified: | 03 Sep 2024 12:42 |
| URI: | https://research.sabanciuniv.edu/id/eprint/49855 |
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