On some Diophantine and Ergodic properties of the Schneider map over function fields

Abstract

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.