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

Abstract Given a subset $$\mathcal {S}\subseteq \mathbb {F}_q[x]$$ S ⊆ F q [ x ] and fixed integers $$n,m\in \mathbb {N}$$ n , m ∈ N , we study the distribution of the smallest denominator $$Q\in \mathcal {S}$$ Q ∈ S for which there exists $$\textbf{P}\in \mathbb {F}_q[x]^m$$ P ∈ F q [ x ] m such that $$\left\| \frac{\textbf{P}}{Q}-\varvec{\alpha }\right\| P Q - α < q - n , where $$\varvec{\alpha }\in x^{-1}\mathbb {F}_q((x^{-1}))^m$$ α ∈ x - 1 F q ( ( x - 1 ) ) m is chosen randomly. We also consider the discrete analogue obtained by fixing a polynomial $$N\in \mathbb {F}_q[x]$$ N ∈ F q [ x ] with $$\textrm{deg}(N)=n$$ deg ( N ) = n and sampling $$\varvec{\alpha }$$ α uniformly from $$\frac{1}{N}\mathbb {F}_q[x]^m$$ 1 N F q