A simple proof of the exponential convergence of the modified Jacobi–Perron algorithm

Abstract

The modified Jacobi–Perron algorithm was shown by Ito etal in 1993 to be strongly convergent (in the sense of Brentjes 1981) almost everywhere with exponential rate. Their proof relies on very complicated computations. We will show that the original article by Podsypanin in 1977 on the modified Jacobi–Perron algorithm almost contains a proof of this convergence with exponential rate. The only ingredients missing in that article are some ergodic-theoretical facts about the transformation generating the approximations. This leads to a very simple proof of the before mentioned exponential convergence in the modified Jacobi–Perron algorithm.