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

Abstract

For an integer $k\geq2$, let $\alpha_k$ 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 $\Sigma(B)=\sum_{b\in B}b, \Sigma(\varnothing)=0$ and $m(B)=\max_{s\in\Z}\#\{C\subseteq B\mid\Sigma(C)=s\},$ and define \[ \beta_k=\sup\left\{c<1\,\middle|\,\lim_{D\to\infty}\Pp\bigl(m(\mathcal A\cap(D^c,D])\geq k\bigr)=1\right\}. \] Ford, Green and Koukoulopoulos proved $\alpha_k\geq\beta_k/(1-\beta_k)$ and conjectured that equality holds for every fixed $k\geq2$. In this paper, we confirm their conjecture. More precisely, for every fixed $a>\beta_k/(1-\beta_k)$, 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.