Linear-fractional invariance of multidimensional continued fractions
type: Item
instance of: scholarly article
title: Linear-fractional invariance of multidimensional continued fractions
zbMATH Open document ID: 1411.11069
author: V. G. Zhuravlev
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-11-02T00:00:00Z
review text: The invariance of the simplex-karyon algorithm for expanding real numbers \(\alpha=(\alpha_1,\alpha_2,\dots,\alpha_d)\) in multidimensional continued fractions under linear-fractional transformations \(\alpha'=(\alpha_1',\alpha_2',\dots,\alpha_d')=U\langle \alpha\rangle\) with matrices \(U\) from the unimodular group \(\mathrm{GL}_{d+1}(\mathbb{Z})\) is established. It is proved that there exists a positive integer \(n_{\alpha,U,\eta^*}\) for which \[ \left| \alpha'_1-\frac{P_{n1}'}{Q_{n1}'} \right|+\dots +\left| \alpha'_d-\frac{P_{nd}'}{Q_{nd}'} \right|\leq \frac{c'}{|Q_{n}'|^{1+\eta^*}} , \quad n \geq n_{\alpha,U,\eta^*}. \] The approximations are the best ones for the new point \(\alpha'\) but with respect the \(s'_n\)-norms, for which the simplices \(s'_n=U \langle\alpha \rangle\), also possessing the contraction property and the minimality property, are chosen as the convex bodies.
reviewed by: Mykhaylo Pahirya
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11B57
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 11Y65
zbMATH DE Number: 6971281
zbMATH Keywords: multidimensional continued fractions
zbMATH Keywords: best approximation
zbMATH Keywords: Farey sums
MaRDI profile type: Publication
full work available at URL: s10958-018-4033-4
OpenAlex ID: W2892235275
cites work: Simplex-karyon algorithm of multidimensional continued fraction expansion
cites work: Q3257189
cites work: Q3284687
cites work: The three-dimensional Poincaré continued fraction algorithm
cites work: Q4501766
cites work: Factor complexity of \(S\)-adic words generated by the Arnoux-Rauzy-Poincaré algorithm
cites work: On some symmetric multidimensional continued fraction algorithms
DOI: 10.1007/S10958-018-4033-4
description: scientific article; zbMATH DE number 6971281