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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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