Minimal Denominators Lying in Subsets of the Ring of Polynomials over a Finite Field
Noy Soffer Aranov
TL;DR
This work extends the minimal-denominator problem from the real setting to the function field context over $\mathbb{F}_q[x]$, proving that for any infinite denominator set $\mathcal{S}$ and any dimension $m$, the discrete distribution of the minimal denominator $d_{N,\mathcal{S}}$ matches the continuous distribution of $\mathrm{deg}_{\min,\mathcal{S}}(\boldsymbol{\alpha},q^{-n})$ for all $n$. It establishes this distributional equivalence via a function-field analogue of Farey fractions and a Ball-Intersection framework in the non-Archimedean setting, with proofs that link discrete and continuous measures. The paper also proves a parallel result for the Q-minimal-denominator distribution $Q_{\min,\mathcal{S}}$, using a separation concept for Farey fractions. In the special case of lacunary semigroups $\mathcal{S}=\{P^d\}$, the authors obtain explicit, tractable distributions for the minimal denominator, highlighting the computational advantages afforded by the function-field setting.
Abstract
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 $\frac{P}{Q}\in B(\boldsymbolα,q^{-n})$, 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 $\boldsymbolα\in \frac{1}{N}\mathbb{F}_q[x]^m$ 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.