论文标题
四分之一功率系列$ \ mathbb {f} _ {3} $的四分力功率系列的Diophantine近似和持续分数扩展
Diophantine approximation and continued fraction expansion for quartic power series over $\mathbb{F}_{3}$
论文作者
论文摘要
尽管罗斯定理指出，所有非理性代数数的非理性度量为2，并且在特征零中的功能字段上相同，但在阳性特征中的函数字段中发现了一些反例。 1949年，马勒（Mahler）在他的关于Diophantine近似\ Cite {M}的基本论文中提出了这一点。似乎，除了特定元素外，作为有界局部商的功率系列，罗斯定理所持。到目前为止，只有一个由Mills和Robbins \ Cite {Mr}发现的带有无界局部商的元素已被认识到拥有此属性。它涉及$ \ mathbb {f} _ {3} $具有持续分数扩展的四分之一的功率系列。 Buck和Robbins \ Cite {Br}明确描述了这种持续的分数扩展，后来又由Lasjaunias \ cite {la2}使用另一种方法更容易。此外，Lasjaunias \ cite {la2}与Roth定理相关的非理性度量的价值。我们将看到，该功率系列包含在一个大型四分之一的功率系列家族中，为此，可以明确给出持续的分数扩展和非理性性措施。此外，我们将研究$ \ mathbb {f} _ {3} $的四分之一功率序列的其他示例的合理近似，我们将扩展Mahler发起的反调查集。
While Roth's theorem states that the irrationality measure of all the irrational algebraic numbers is 2, and the same holds true over function fields in characteristic zero, some counter-examples were found over function fields in positive characteristic. This was put forward first by Mahler in 1949, in his fundamental paper on Diophantine approximation \cite{M}. It seems that, except for particular elements, as power series with bounded partial quotients, Roth's theorem holds. Until now, only one element, with unbounded partial quotients, discovered by Mills and Robbins \cite{MR} in 1986, has been recognized having this property. It concerns a quartic power series over $\mathbb{F}_{3}$ having a continued fraction expansion with remarkable pattern. This continued fraction expansion was explicitly described by Buck and Robbins \cite{BR}, and later by Lasjaunias \cite{LA2} who used another method somewhat easier. Furthermore, Lasjaunias \cite{LA2} improve the value of its irrationality measure in relation with Roth's theorem. We will see that this power series is included in a large quartic power series family, for which the continued fraction expansion and the irrationality measure can be explicitly given. Moreover, we will study the rational approximation of other examples of quartic power series over $\mathbb{F}_{3}$ and we will extend the set of counter-examples initiated by Mahler.
