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

Huayang Chen; Alan Haynes

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arXiv · 2109.12668

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

Huayang Chen·Alan Haynes

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

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Huayang Chen, Alan Haynes. 2021-09-26. Expected value of the smallest denominator in a random interval of fixed radius. https://arxiv.org/abs/2109.12668

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