On the Minimal Denominator Problem in Function Fields

Abstract

We study the minimal denominator problem in function fields. In particular, we compute the probability distribution function of the the random variable which returns the degree of the smallest denominator $Q$, for which the ball of a fixed radius around a point contains a rational function of the form $\frac{P}{Q}$. Moreover, we discuss the distribution of the random variable which returns the denominator of minimal degree, as well as higher dimensional and $P$-adic generalizations. This can be viewed as a function field generalization of a paper by Chen and Haynes.