Quadratic approximation in $$\mathbb{F}_q(\!(T^{-1})\!)$$.(English)Zbl 1381.11058

Let $$p$$ be a prime and $$q$$ be a power of $$p$$. Denote by $$\mathbb{F}_q$$ the finite field of $$q$$ elements, $$\mathbb{F}_q[T]$$ the ring of polynomials over $$\mathbb{F}_q$$, $$\mathbb{F}_q(T)$$ the field of rational function over $$\mathbb{F}_q$$, and $$\mathbb{F}_q((T^{-1}))$$ the field of Laurent series over $$\mathbb{F}_q$$. If $$\xi \in \mathbb{F}_q((T^{-1}))$$, one can write $$\xi =\sum_{n=N}^{\infty} a_n T^{-n},$$ where $$N \in \mathbb{Z}, a_n \in \mathbb{F}_q,$$ and $$a_N\neq 0.$$ The author regards elements of $$(\mathbb{F}_q[T])[X]$$ as polynomials in $$X$$. The maximal of absolute values of the coefficients of $$P(x) \in (\mathbb{F}_q[T])[X]$$ is denoted by $$H(P)$$. Let $w_n(\xi,H)=\min \{ |P(\xi)|\mid P(X)\in (\mathbb{F}_q[T])[X], H(P) \leq H, \deg_X P\leq n, P(\xi)\neq 0 \},$
$w^*_n(\xi,H) =\min\{|\xi-\alpha|\mid \alpha \in \overline{\mathbb{F}_q(T)}, H(\alpha) \leq H, \deg \alpha \leq n, \alpha\neq \xi \}.$ The Diophantine exponents $$w_n$$ and $$w^*_n$$ are defined by $w_n(\xi)=\limsup_{H \to \infty} \frac{-\log w_n(\xi,H)}{\log H}, \qquad w^*_n(\xi)=\limsup_{H \to \infty} \frac{-\log H w_n^*(\xi,H)}{\log H}.$ The difference of the Diophantine exponents $$w_2$$ and $$w_2^*$$, using continued fractions, is studied in this article. It is shown that the range of function $$w_2-w_2^*$$ is exactly the closed interval $$[0,1]$$. An upper bound of the exponent $$w_2$$ of continued fractions with low complexity partial quotients is estimated.

MSC:

 11J61 Approximation in non-Archimedean valuations 11J70 Continued fractions and generalizations 11J82 Measures of irrationality and of transcendence
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