Poisson laws and arithmetic clusters for rare multiplicative patterns
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 $L^2$ densities, we approximate the field of hits of site-dependent targets of Haar mass $O(N^{-1})$ on deterministic masks in $[2,CN]$. The target retains the true prime-site intensities and uses Haar at composite sites, even when the total prime contribution diverges. The exponent $2$ is sharp in the uniform $L^q$ scale for this target and class of masks. Below this threshold, even…
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