Minimal Denominators Lying in Subsets of the Ring of Polynomials over a Finite Field

Abstract

Given a subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$ and fixed integers $n,m\in \mathbb{N}$, we study the distribution of the smallest denominator $Q\in \mathcal{S}$ for which there exists $\mathbf{P}\in \mathbb{F}_q[x]^m$ such that $\left\Vert\frac{\mathbf{P}}{Q}-\boldsymbol{\alpha}\right\Vert<q^{-n}$, where $\boldsymbol{\alpha}\in x^{-1}\mathbb{F}_q((x^{-1}))^m$ is chosen randomly. We also consider the discrete analogue obtained by fixing a polynomial $N\in \mathbb{F}_q[x]$ with $\deg(N)=n$ and sampling $\boldsymbol{\alpha}$ uniformly from $\frac{1}{N}\mathbb{F}_q[x]^m$. We prove that for any infinite subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$, for every $n\in \mathbb{N}$ and every dimension $m$, the probability distributions of these two random variables coincide. This result is significantly stronger than the corresponding statement in the real setting, where Balazard and Martin showed that the averages of the discrete and continuous smallest denominator functions are asymptotically close.