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

Given a subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$ and fixed $n,m\in \mathbb{N}$, one can study the distribution of the value of the smallest denominator $Q\in \mathcal{S}$, for which there exists $\mathbf{P}\in \mathbb{F}_q[x]^m$ such that $\frac{P}{Q}\in B(\boldsymbol{\alpha},q^{-n})$, where $Q\in \mathcal{S}$. On the other hand, one can study the discrete analogue, when $N\in \mathbb{F}_q[x]$ is a polynomial with $\deg(N)=n$ and $\boldsymbol{\alpha}\in \frac{1}{N}\mathbb{F}_q[x]^m$ as a discrete probability distribution function. We prove that for any infinite subset $\mathcal{S}\subseteq \mathbb{F}_q[x]$, for any $n\in \mathbb{N}$, and for any dimension $m$, the probability distribution functions of both these random variables are equal to one another. This is significantly stronger than the real setting, where Balazard and Martin proved that these functions have asymptotically close averages, when there are no restrictions on the denominators.