Attenuated Poisson Dirichlet approximations for divisibility configurations

Victor Bernal Ramirez; David Torres-Flores; Arturo Jaramillo

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arXiv · 2606.30349

Attenuated Poisson Dirichlet approximations for divisibility configurations

Victor Bernal Ramirez·David Torres-Flores·Arturo Jaramillo

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Abstract

We study the point process formed by the normalized logarithms of the distinct prime factors of a harmonic random sample. We prove a quantitative convergence result, in a Wasserstein-type metric over decreasing sequences, toward the atom sequence of a Dickman Poisson cloud conditioned to have total mass at most one, equivalently a uniformly attenuated Poisson-Dirichlet law. The proof is based on the conditioned geometric representation of harmonic samples, a Poisson approximation chain for the associated point processes, monotone couplings of Poisson point processes, and Kolmogorov estimates for the Dickman approximation of weighted geometric sums.

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Victor Bernal Ramirez, David Torres-Flores, Arturo Jaramillo. 2026-06-29. Attenuated Poisson Dirichlet approximations for divisibility configurations. https://arxiv.org/abs/2606.30349

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