Year: 2026
Type: preprint
Abstract: For $k\geq2$, let $α_k$ be the supremum of the exponents $a$ for which almost every integer $n$ has $k$ distinct divisors in a multiplicative interval of relative length $(\log n)^{-a}$. Select each positive integer $i$ independently with probability $1/i$, forming a random set $\mathbf A$, and let $β_k$ be the supremum of the $c<1$ for which, with probability tending to one as $D\to\infty$, the set $\mathbf A\cap[D^c,D]$ has $k$ distinct subsets with the same sum. We prove that $α_k=β_k/(1-β_k)$, resolving a conjecture of Ford, Green and Koukoulopoulos [Invent. Math. 232 (2023), 1027--1160]. We also prove that their weak and strict entropy thresholds coincide. The proof combines flag refinement and entropy concavity with an upper bound for approximate subset sums that is uniform in arbitrary translations. A model with independent geometric prime exponents then transfers this bound to divisors.
Source: arXiv (Cornell University)
Authors: Hou, Jianfeng, Zhao, Hongbin
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214088653