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 $\delta \to 0$, to $(16/{\pi }^2){\delta }^{-1/2}.$

9 pages, streamlined proof, radius $\delta$ from previous version replaced by $\delta /2$

References in corpus (1)

• Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps