On the Minimal Denominator Problem in Function Fields

TL;DR

This paper investigates the distribution of minimal denominators in function fields, providing probabilistic insights and generalizations, extending previous work to higher dimensions and $P$-adic contexts.

Contribution

It computes the probability distribution of minimal denominators in function fields and explores their higher-dimensional and $P$-adic generalizations, offering a comprehensive analysis.

Findings

Distribution of minimal denominators characterized

Probabilistic models for rational functions established

Extensions to higher dimensions and $P$-adic cases discussed

Abstract

We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the the random variable which returns the degree of the smallest denominator $Q$, for which the ball of a fixed radius around a point contains a rational function of the form $\frac{P}{Q}$. Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and $P$-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.