On Schweiger's problems on fully subtractive algorithms
type: Item
instance of: scholarly article
title: On Schweiger's problems on fully subtractive algorithms
zbMATH Open document ID: 1273.11119
author: Hitoshi Nakada
author: Robbert Fokkink
author: Cor Kraaikamp
published in: Israel Journal of Mathematics
publication date: 2012-11-19T00:00:00Z
review text: Let \(x_{1}<x_{2}<\ldots <x_{n}\;\) be real numbers that are rationally independent. For \(\;1\leq a<n\) \ \ consider the fully substractive algorithm \[ \tau:(x_{1},\ldots ,x_{a},x_{a+1},\ldots ,x_{n}\longrightarrow \text{sort}\;(x_{1},\ldots ,x_{a},x_{a+1}-x_{a},\ldots ,x_{n}-x_{a}). \] For \(a=1\), it appears in percolation theory and was successively studied by R. Meester (1989), R. Meester and T. Nowicki (1989) and C. Kraaikamp and R. Meester (1995). For arbitrary \(a\), the algorithm was further associated with multidimensional continued fractions. In his monograph [Multidimensional continued fractions. Oxford: Oxford University Press. (2000; Zbl 0981.11029)], see especially page 81, \textit{F. Schweiger} presented the following two conjectures. 1. With respect to Lebesgue measure, for almost every \(n\)-tuple \(x,\) there exists an iterate \(x^{m}=\tau ^{m}\left( x\right) \) such that\ the sum of the first \(a+1\) coordinates of \(x^{m}\) is smaller than its \(a+2\)-nd coordinate. 2. \(\tau \) is ergodic with respect to Lebesgue measure. The present authors show that 1 is true while 2 is false.
reviewed by: Marius Iosifescu
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 37A45
zbMATH DE Number: 6108401
zbMATH Keywords: multidimensional continued fraction
zbMATH Keywords: ergodic theory
MaRDI profile type: Publication
full work available at URL: s11856-011-0140-3
OpenAlex ID: W2021300673
cites work: Ergodic properties of a dynamical system arising from percolation theory
cites work: An algorithm for calculating critical probabilities and percolation functions in percolation models defined by rotations
cites work: Infinite clusters and critical values in two-dimensional circle percolation
cites work: Q3666965
cites work: Q4036525
cites work: Q4501766
cites work: Fine and Wilf words for any periods
DOI: 10.1007/S11856-011-0140-3
Recommended article: Brun meets Selmer
Recommended article: Ergodic properties of a dynamical system arising from percolation theory
Recommended article: Absorbing sets of homogeneous subtractive algorithms
Recommended article: Lebesgue Ergodicity of a Dissipative Subtractive Algorithm
Recommended article: Q4441099
Recommended article: Q3348039
Recommended article: Q4896208
Recommended article: EXPONENTIALLY STRONG CONVERGENCE OF NON-CLASSICAL MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHMS
Recommended article: The convergence of the generalised Selmer algorithm
Recommended article: Q3480145
description: scientific article; zbMATH DE number 6108401