Quadratic approximation in $\mathbb{F}_q ((T^{-1}))$

TL;DR

This paper investigates quadratic approximation in Laurent series over finite fields, characterizing the range of Diophantine exponents and providing bounds for continued fraction exponents with low complexity partial quotients.

Contribution

It precisely determines the range of the difference between Diophantine exponents for quadratic approximation over finite fields.

Findings

The range of w_2 - w_2^* is exactly [0,1].

An upper bound for w_2 is established for continued fractions with low complexity partial quotients.

The study advances understanding of Diophantine approximation in finite field Laurent series.

Abstract

In this paper, we study Diophantine exponents $w_n$ and $w_n ^{*}$ for Laurent series over a finite field. Especially, we deal with the case $n=2$, that is, quadratic approximation. We first show that the range of the function $w_2-w_2 ^{*}$ is exactly the closed interval $[0,1]$. Next, we estimate an upper bound of the exponent $w_2$ of continued fractions with low complexity partial quotients.