发表机构
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Ford-Green-Koukoulopoulos猜想：几乎每个整数的除数分布与随机集合的等和子集存在精确关系，并证明弱熵阈值与严格熵阈值一致，方法结合旗细化、熵凹性和均匀平移的近似子集和上界。
AI 中文摘要
对于$k\geq2$，设$\alpha_k$为指数$a$的上确界，使得几乎所有整数$n$在相对长度为$(\log n)^{-a}$的乘法区间内都有$k$个不同的除数。独立地以概率$1/i$选择每个正整数$i$，形成随机集合$\mathbf A$，并设$\beta_k$为$c<1$的上确界，使得当$D\to\infty$时，概率趋于1，集合$\mathbf A\cap[D^c,D]$具有$k$个和相同的不同子集。我们证明$\alpha_k=\beta_k/(1-\beta_k)$，解决了Ford、Green和Koukoulopoulos的猜想[Invent. Math. 232 (2023), 1027--1160]。我们还证明了他们的弱熵阈值和严格熵阈值一致。证明结合了旗细化与熵凹性，以及对任意平移一致的近似子集和的上界。一个具有独立几何素数指数的模型随后将此上界转移到除数上。
英文摘要
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.
Comments12 pages
