A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields
In [Acta Arith. 150, No. 2, 129--142 (2011; Zbl 1248.11050)], \textit{D. H. Kim} and \textit{H. Nakada} proved an analogue of \textit{J. Kurzweil}'s theorem [Stud. Math. 15, 84--112 (1955; Zbl 0066.03702)] for \textit{inhomogeneous Diophantine approximation} of \textit{formal Laurent series} over finite fields, with use of continued fraction theory. In this article, the authors generalize this result to \textit{simultaneous Diophantine approximation} with a proof of different nature. Let \(\mathbb F_q\) be the finite field with \(q\) elements, \(\mathbb F_q[T]\) the polynomial ring and \(\mathbb F_q((T^{-1})):=\{ f= a_nT^n+a_{n-1} T^{n-1}+\dots : a_i\in \mathbb F_q, n\in \mathbb Z\}\) the field of formal Laurent series. For a formal Laurent series \(f=a_nT^n+a_{n-1}T^{n-1}+\dots \), we define its fractional part by \(\{ f\} =a_{-1}T^{-1}+a_{-2}T^{-2}+\dots,\) and its valuation by \(|f|=q^{\deg f}\) where \(\deg f\) is the generalized degree function. Let us define \(\mathbb L :=\{ f\in \mathbb F_q((T^{-1})) : |f|<1\}\). Fix two non-negative integers \(r\) and \(s\). Denote by \(\mathbb F_q[T]^r\) the \(r\)-th fold cartesian product of \(\mathbb F_q[T]\) and by \(\mathbb F_q((T^{-1}))^r\) the \(r\)-th dimensional vector space over \(\mathbb F_q((T^{-1}))\). Let \(\mathbf f=(f_1,\dots,f_r)\in\mathbb F_q((T^{-1}))^r\) be a vector. Then define its fractional part by \(\{\mathbf f\}:= (\{f_1\},\dots,\{f_r\})\) and its \textit{valuation } \(\| \mathbf f\| =q^{\deg \mathbf f }=\max _{1\leq i\leq r}|f_i|\), where \(\deg \mathbf f=\max_{1\leq i\leq r} \deg f_i\). Note that \(\|\cdot\| \) satisfies the \textit{ultra-metric} property. Let \(\mathbb L^r\) denote the \(r\)-th fold Cartesian product of \(\mathbb L\) equipped with the \textit{product measure} of \(\mathbb L\). For \(A\in \mathbb L^{r\times s}\) and \(\mathbf g\in \mathbb L^s\), let us consider \[ \|\{ \mathbf qA\} -\mathbf g\| < \frac{1}{q^{\lfloor\frac{nr}{s}\rfloor+l_n}}, \mathbf q\in \mathbb F_q[T]^r, \deg \mathbf q=n, \tag{2} \] where \(\l_n\) is a sequence of non-negative integers. In this article, the authors assume that \(A\) is fixed and \(\mathbf g\) is random. \(A\in \mathbb L^{r\times s}\) is said \textit{badly approximable } if there exists a \(c\in\mathbb N\) such that for all \(\mathbf q\in \mathbb F_q[T]^r\) with \(\deg \mathbf q=n\), we have \(\|\{\mathbf qA\}\|\geq \frac{1}{q^{\lfloor\frac{nr}{s}\rfloor+c}}.\) Let us define the set \[ \begin{multlined} W_{r,s}=\Big\{ A\in \mathbb L^{r\times s} :\;\forall\;l_n\text{ with } \sum_{n=0}^\infty q^{-l_n s}=\infty, \\ (2) \text{ has infinitely many solutions for almost all } \mathbf g\Big\}.\end{multlined} \] The authors prove the following: We have \(W_{r,s}= \{A\in \mathbb L^{r\times s} : A \text{ is badly approximable}\}.\)
scientific article; zbMATH DE number 6110665
scientific article