AI 中文总结
该论文研究函数域上Schneider映射定义的连分数，证明其Laurent级数域元素的连分数展开存在唯一，确立相关丢番图逼近与遍历性质，确定最难逼近元素集的Hausdorff维数，为函数域数论与动力系统提供新结果。
AI 中文摘要
我们引入并研究多项式环上由类Schneider映射定义的连分数，这类映射与任意次的固定多项式相关联。特别地，我们证明了Laurent级数域中每个元素的连分数展开都存在且唯一；随后确定了对应收敛分数的精确丢番图逼近性质；还研究了该映射的动力学性质，证明Haar测度是不变且遍历的；作为算术推论，我们确定了几乎所有元素展开的数字的渐近和；最后，我们识别了该框架下最难逼近的元素集合，并计算了其Hausdorff维数，表明它是正维数的分形集。
英文摘要
We introduce and study continued fractions defined by Schneider-like maps over polynomial rings, where the maps are associated with a fixed polynomial of arbitrary degree. In particular, we prove the existence and uniqueness of the continued fraction expansion for every element of the field of Laurent series. We then establish precise Diophantine approximation properties of the corresponding convergents. We also study the dynamical aspects of the underlying map, proving that the Haar measure is invariant and ergodic. As an arithmetic consequence, we determine the asymptotic sum of the digits of the expansion for almost every element. Finally, we identify the set of elements that are worst approximable in this framework and compute its Hausdorff dimension, showing that it is a fractal set of positive dimension.
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