Minimal Denominators Lying in Subsets of the Ring of Polynomials over a Finite Field
This provides a foundational result in number theory and coding theory for researchers studying approximations in finite fields, with potential applications in cryptography and error-correcting codes.
The paper tackles the problem of analyzing the distribution of minimal denominators in polynomial rings over finite fields, proving that for any infinite subset, the probability distributions of discrete and continuous smallest denominator functions coincide exactly for all dimensions, which is stronger than the asymptotic result known in the real setting.
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 $\left\Vert\frac{\mathbf{P}}{Q}-\boldsymbolα\right\Vert<q^{-n}$, where $\boldsymbolα\in x^{-1}\mathbb{F}_q((x^{-1}))^m$ is chosen randomly. We also consider the discrete analogue obtained by fixing a polynomial $N\in \mathbb{F}_q[x]$ with $°(N)=n$ and sampling $\boldsymbolα$ 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.