Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture
Abstract
For an integer k≥2, let αk be the supremum of the real numbers a for which almost every integer n≥2 has divisors d1<⋯<dk| n satisfying dk≤ d1(1+(log n)−a). Let 𝒜⊆N be the logarithmic random set in which the events m∈𝒜 are mutually independent and Pp(m∈𝒜)=1/m for every m≥1. For a finite set B⊆N, write Σ(B)=∑b∈ Bb, Σ(∅)=0 and m(B)=maxs∈Z#{C⊆ B|Σ(C)=s}, and define βk=sup{c<1 middle| limD→∞Pp(m(𝒜∩(Dc,D])≥ k)=1}. Ford, Green and Koukoulopoulos proved αk≥βk/(1−βk) and conjectured that equality holds for every fixed k≥2. In this paper, we confirm their conjecture. More precisely, for every fixed a>βk/(1−βk), almost every integer n≥2 has no divisors d1<⋯<dk| n satisfying dk≤ d1(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.
Related
- Extension of solvable coverings across horizontal divisors over the integers
- Equal subset sums and close divisors
- Generic vanishing conjecture for divisors
- Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture
- Consecutive integers coalesce with density one under the Collatz map: a formally verified proof of a conjecture of Gao
BibTeX
@misc{mao2026close,
title = {{Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture}},
author = {Yaping Mao and Yanyan Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22296},
eprint = {2609.22296},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22296},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Close Divisors of Typical Integers:The Ford--Green--Koukoulopoulos Conjecture
AU - Yaping Mao
AU - Yanyan Song
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.22296
UR - https://arxiv.org/abs/2609.22296
AB - For an integer k≥2, let α_k be the supremum of the real numbers a for which almost every integer n≥2 has divisors d_1<⋯<d_k| n satisfying d_k≤ d_1(1+(log n)^−a). Let 𝒜⊆N be the logarithmic random set in which the events m∈𝒜 are mutually independent and Pp(m∈𝒜)=1/m for every m≥1. For a finite set B⊆N, write Σ(B)=∑_b∈ Bb, Σ(∅)=0 and m(B)=max_s∈Z#{C⊆ B|Σ(C)=s}, and define β_k=sup{c<1 middle| lim_D→∞Pp(m(𝒜∩(D^c,D])≥ k)=1}. Ford, Green and Koukoulopoulos proved α_k≥β_k/(1−β_k) and conjectured that equality holds for every fixed k≥2. In this paper, we confirm their conjecture. More precisely, for every fixed a>β_k/(1−β_k), almost every integer n≥2 has no divisors d_1<⋯<d_k| n satisfying d_k≤ 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.
ER -