Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 8 min ago
PreprintCC BY 4.0arXiv

A Resolution of the de Bruijn--Erdős Consecutive-Gap Problem

  • Samuel Korsky
Open at sourcePDFDOICite

Abstract

Let (xn)n≥1 be a sequence of distinct points on the unit circle. An r-span is the total length of r consecutive gaps determined by the inserted points. Write Mn(r) and mn(r) for the largest and smallest r-spans after the first n insertions. We prove that there is an absolute constant c>0 such that, for every sufficiently large r, limsupn→∞(nMn(r)−r) ≥ c√(log r), limsupn→∞(r−nmn(r)) ≥ c√(log r), and limsupn→∞Mn(r)/mn(r) ≥ 1+(log r)/100r. Thus all three asymptotic conjectures made by de Bruijn and Erdős in 1949 are resolved. The ratio bound matches the upper bound of Clément and Steinerberger up to an absolute constant and answers a question of Brethouwer. The proofs compare interval counts at nearby times. Pointwise control leads to a one-dimensional sequence-discrepancy argument for the ratio, while averaged control and Halász's planar L1 discrepancy theorem give the two one-sided conclusions.

Related

  1. A finite victory over de Bruijn-Erdős in interval discrepancy
    Jared DeLeo, Owen Henderschedt, Chris Wells
    arXivPreprintCC BY-NC-SA 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. The Erdős-Gallai bound for consecutive even cycle lengths
    Yaobin Chen, Hong Liu, Xia Wang +2
    arXivPreprintCC BY-NC-SA 4.0Full textCombinatorics & discrete mathematicsarXiv
  3. Linear dependencies, polynomial factors in the Duke–Erdős forbidden sunflower problem – CORRIGENDUM
    Andrey Kupavskii, Fedor Noskov
    Forum of Mathematics, SigmaPeer-reviewedCC BY 4.0General mathematicsCrossref
  4. On the Erdős--Ginzburg--Ziv Problem for ℤnr
    Dongchun Han, Hanbin Zhang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. On a problem of Erdős and Nathanson related to minimal asymptotic bases of order h
    Shi-Qiang Chen, Quan-Hui Yang
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{korsky2026resolution,
  title = {{A Resolution of the de Bruijn--Erdős Consecutive-Gap Problem}},
  author = {Samuel Korsky},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07196},
  eprint = {2609.07196},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07196},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Resolution of the de Bruijn--Erdős Consecutive-Gap Problem
AU  - Samuel Korsky
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.07196
UR  - https://arxiv.org/abs/2609.07196
AB  - Let (x_n)_n≥1 be a sequence of distinct points on the unit circle. An r-span is the total length of r consecutive gaps determined by the inserted points. Write M_n^(r) and m_n^(r) for the largest and smallest r-spans after the first n insertions. We prove that there is an absolute constant c>0 such that, for every sufficiently large r, limsup_n→∞(nM_n^(r)−r) ≥ c√(log r), limsup_n→∞(r−nm_n^(r)) ≥ c√(log r), and limsup_n→∞M_n^(r)/m_n^(r) ≥ 1+(log r)/100r. Thus all three asymptotic conjectures made by de Bruijn and Erdős in 1949 are resolved. The ratio bound matches the upper bound of Clément and Steinerberger up to an absolute constant and answers a question of Brethouwer. The proofs compare interval counts at nearby times. Pointwise control leads to a one-dimensional sequence-discrepancy argument for the ratio, while averaged control and Halász's planar L^1 discrepancy theorem give the two one-sided conclusions.
ER  -