Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture

Abstract

For an integer $k\ge 2$, let ${\alpha }_k$ be the supremum of the real numbers $a$ for which almost every integer $n\ge 2$ has divisors $d_1<\cdots <d_k\mid n$ satisfying $d_k\le d_1(1+(\log n{)}^{-a}).$ Let $\mathcal{A}\subseteq \mathbb{N}$ 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 $m\ge 1$. For a finite set $B\subseteq \mathbb{N}$, write $\Sigma (B)={\sum }_{b\in B}b,\Sigma (\varnothing )=0$ and $m(B)={\max }_{s\in \mathbb{Z}}\#\{C\subseteq B\mid \Sigma (C)=s\},$ 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 ${\alpha }_k\ge {\beta }_k/(1-{\beta }_k)$ and conjectured that equality holds for every fixed $k\ge 2$. In this paper, we confirm their conjecture. More precisely, for every fixed $a>{\beta }_k/(1-{\beta }_k)$, almost every integer $n\ge 2$ has no divisors $d_1<\cdots <d_k\mid n$ satisfying $d_k\le d_1(1+(\log n{)}^{-a})$. 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.