On quadratic approximation for hyperquadratic continued fractions

Abstract

We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.

Mahler’s and Koksma’s Classifications in Fields of Power Series

“…For , Ooto [19] proved the existence of in for which . His strategy, inspired by [6], was to use continued fractions to construct power series with and sufficiently large to ensure that, for small (in terms of ) values of n , we have Very recently, Ayadi and Ooto [2] answered a question of Ooto [20, Problem 2.2] by proving, for given and , the existence of algebraic power series in for which .…”

“…We obtain (2.1) by taking Wirsing's argumentation [28]. Let n ≥ 2 be an integer, and let ξ be a power series which is either transcendental, or algebraic of degree > n. Let ε > 0, and set w = w n (ξ)(1 + ε) 2 . Let i 1 , .…”

“…Very recently, Ayadi and Ooto [2] answered a question of Ooto [20, Problem 2.2] by proving, for given n ≥ 2 and q ≥ 4, the existence of algebraic power series ξ in F q ((T −1 )) for which w * n (ξ) < w n (ξ).…”

Mahler's and Koksma's classifications in fields of power series

“…= w n (ξ). Very recently, Ayadi and Ooto [2] answered a question of Ooto [21, Problem 2.2] by proving, for given n ≥ 2 and q ≥ 4, the existence of algebraic power series ξ in F q ((T −1 )) for which w * n (ξ) < w n (ξ). Problem 6.2.…”

A note on the length of some finite continued fractions

“…The analogue of this continued fraction in the function field case allows us to describe the continued fraction for many algebraic irrational power series, see for example [2,4]. The great interest of these continued fractions will give us the opportunity to study their analogues in the real number case.…”