Expected value of the smallest denominator in a random interval of fixed radius

Abstract

We compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius $\delta/2$. As an application, we prove that the expected value of the smallest denominator is asymptotic, as $\delta\rightarrow 0$, to $(16/\pi^2)\delta^{-1/2}.$