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

TL;DR

This paper calculates the probability distribution of the smallest denominator in a random interval and shows its expected value grows asymptotically as the interval radius shrinks.

Contribution

It derives the probability mass function for the smallest denominator in a random interval and establishes its asymptotic expected value as the interval radius approaches zero.

Findings

Probability mass function of the smallest denominator computed.

Expected value asymptotic to (16/π^2)δ^{-1/2} as δ→0.

Provides analytical tools for understanding denominators in random intervals.

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