On Schweiger's problems on fully subtractive algorithms (Q1758958)

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.