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Poisson laws and arithmetic clusters for rare multiplicative patterns

Primarily AI-generated textHuman understanding: no partsmath.PR — Probabilitymath.NT — Number Theory

Contributed by Brice Pouly ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

We determine when rare observations of completely multiplicative functions have a Poisson law and when arithmetic relations force clusters. For independent prime values on the circle with uniformly bounded L2L^2 densities, we approximate the field of hits of site-dependent targets of Haar mass O(N−1)O(N^{-1}) on deterministic masks in [2,CN][2,CN]. The target retains the true prime-site intensities and uses Haar at composite sites, even when the total prime contribution diverges. The exponent 22 is sharp in the uniform LqL^q scale for this target and class of masks. Below this threshold, even correct prime intensities can fail at site resolution while the total count remains asymptotically Poisson. A fixed finite exceptional prime environment, possibly atomic and internally dependent, instead produces a conditional cluster law together with an exact environment-determined configuration. With Haar outside that environment, its integer relation lattice determines the compound-Poisson limit of the corrected prefix count and characterizes simple Poisson convergence. Further applications treat rational observations on sparse moving supports, permanent bounded prime biases, and - using a binary arithmetic relation profile - long Borel patterns in a finite abelian group times a torus. Classical marked Chen-Stein comparison provides a common interface. A separate deterministic-sequence interface gives correlation identities, transfer criteria and a resolution obstruction, without proving cancellation for Liouville.

Provenance statement

The following declaration describes the research and the published source edition: The research was organized and directed by the author with substantial assistance from GPT-6 Astra (OpenAI), the only language-model system used for this article. It was used to explore candidate lemmas and proof routes, draft, translate and reorganize the manuscript, write diagnostic programs, check references and build artifacts, and conduct adversarial reviews through separate instances. The author determines the claims and scope and accepts responsibility for the arguments, references and any remaining errors. AI-assisted checks are not independent human peer review. The mathematical claims rest on the printed arguments and the stated external results, not on model output or the success of finite diagnostic tests. For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review. The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.22877400. At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026