Metrical properties for continued fractions of formal Laurent series

Abstract

Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let $A_n(x)$ be the $n$th partial quotient of the continued fraction expansion of $x$ in the field of formal Laurent series. We consider the sets of $x$ such that $°A_{n+1}(x)+\cdots +°A_{n+k}(x)\ge \Phi (n)$ holds for infinitely many $n$ and for all $n$ respectively, where $k\ge 1$ is an integer and $\Phi (n)$ is a positive function defined on $\mathbb{N}$. We determine the size of these sets in terms of Haar measure and Hausdorff dimension.

arXiv comment22 pages

References in corpus (5)

• Hausdorff measure of sets of Dirichlet non-improvable numbers
• Hausdorff dimension of a set in the theory of continued fractions
• Dirichlet's Theorem in function fields
• On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series
• The sets of Dirichlet non-improvable numbers vs well-approximable numbers