发表机构
School of Mathematics and Statistics, Qinghai Normal University（青海师范大学数学与统计学院）
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：对每个固定的 $k\geq2$，典型整数的接近除数指数 $\alpha_k$ 恰等于 $\beta_k/(1-\beta_k)$，其中 $\beta_k$ 由对数随机集合的子集和多重性定义。
AI 中文摘要
对于整数 $k\geq2$，设 $\alpha_k$ 为实数 $a$ 的上确界，使得几乎所有整数 $n\geq2$ 都有除数 $d_1<\cdots<d_k\mid n$ 满足 $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr)$。设 $\mathcal A\subseteq\N$ 为对数随机集合，其中事件 $m\in\mathcal A$ 相互独立，且对每个 $m\geq1$ 有 $\Pp(m\in\mathcal A)=1/m$。对于有限集合 $B\subseteq\N$，记 $\Sigma(B)=\sum_{b\in B}b$，$\Sigma(\varnothing)=0$，$m(B)=\max_{s\in\Z} \\#\{C\subseteq B\mid\Sigma(C)=s\}$，并定义 \\[ \beta_k= \sup\left\{ c<1\\,\middle|\\, \lim_{D\to\infty} \Pp\bigl(m(\mathcal A\cap(D^c,D])\geq k\bigr)=1 \right\}. \\] Ford、Green 和 Koukoulopoulos 证明了 $\alpha_k\geq\beta_k/(1-\beta_k)$ 并猜想对每个固定的 $k\geq2$ 等式成立。本文证明了该猜想。更精确地，对每个固定的 $a>\beta_k/(1-\beta_k)$，几乎所有整数 $n\geq2$ 都没有满足 $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr)$ 的除数 $d_1<\cdots<d_k\mid n$。
英文摘要
For an integer $k\geq2$, let $α_k$ be the supremum of the real numbers $a$ for which almost every integer $n\geq2$ has divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr).$ Let $\mathcal A\subseteq\N$ be the logarithmic random set in which the events $m\in\mathcal A$ are mutually independent and $\Pp(m\in\mathcal A)=1/m$ for every $m\geq1$. For a finite set $B\subseteq\N$, write $Σ(B)=\sum_{b\in B}b, Σ(\varnothing)=0$ and $m(B)=\max_{s\in\Z}\#\{C\subseteq B\midΣ(C)=s\},$ and define \[ β_k=\sup\left\{c<1\,\middle|\,\lim_{D\to\infty}\Pp\bigl(m(\mathcal A\cap(D^c,D])\geq k\bigr)=1\right\}. \] Ford, Green and Koukoulopoulos proved $α_k\geqβ_k/(1-β_k)$ and conjectured that equality holds for every fixed $k\geq2$. In this paper, we confirm their conjecture. More precisely, for every fixed $a>β_k/(1-β_k)$, almost every integer $n\geq2$ has no divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr)$. We also correct local errors in their paper [\emph{Invent. Math.} 232 (2023), 1027--1160], concerning the finite-subflag reduction, the residual-sum count, the moment estimate and the lattice adjustment. These corrections preserve the entropy-threshold comparison used in our proof.
