Strong convergence of additive Multidimensional Continued Fraction algorithms

Abstract

1. Statements of results. By the paper of Lagarias [4], the metrical convergence of algorithms of simultaneous Diophantine approximation has been clearly formulated as a problem of ergodic theory and dynamical systems. Since then, much of the study was devoted to variants of the Jacobi-Perron algorithm, which is a direct generalisation of the continued fraction algorithm. There is another class of algorithms which includes the algorithms of Selmer and Brun. This class appears to be more practical since the operations of approximation consist of addition (subtraction) instead of multiplication as in the case of the Jacobi-Perron type algorithms. In this paper, we prove the almost everywhere strong convergence of a class of two-dimensional algorithms of additive type (Corollary 2). The problem of strong convergence concerns the speed of convergence of Diophantine quantities q n x − p n , which typically attenuate oscillating as n goes to infinity. For future reference, we formulate our criterion for arbitrary finite-dimensional algorithms (Theorem 1).Metric properties of Selmer's algorithm have already been studied by Schweiger [5]. Schweiger reduces Selmer's algorithm by the jump transformation to Baldwin's one, the ergodic properties of which are already established. However, his argument covers one particular sequence of fractions of approximation. Our method covers all the fractions given by the algorithm, which is complementary to Schweiger [5] in this respect, and as a consequence, we obtain information on the second Lyapunov exponent of the system. For a proof of the strong convergence of two-dimensional Brun's algorithm for the multiplicative case (cf. Remark 1

Symbolic coding of linear complexity for generic translations of the torus, using continued fractions

“…Moreover we know that Cassaigne algorithm is conjugated to Selmer algorithm [14], thus we deduce the ergodicity of this measure. It is well known that for the Selmer algorithm the second Lyapunov exponent is strictly negative, with x → codim(E 2 (x)) µ-almost surely constant to 1, see [30]. Thus by conjugation we deduce θ 2 (F, µ) < 0 and codim(E 2 (x)) = 1 for µ-almost every x ∈ X.…”

On $p$-adic multidimensional continued fractions

“…In the previous theorems, we have seen the weak convergence of a MCF. Furthermore, we can show that in the case of convergent p-adic MCFs we also have strong convergence (for the definition of strong convergence in the context of continued fractions see, e.g., [24]). Let us define…”