Korsky: A resolution of the de Bruijn–Erdős consecutive-gap problem

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theorem_1_1: For all large r and every sequence of distinct points on the circle, the upper limits of n M_n − r and of r − n m_n are at least c root log r and the upper limit of M_n/m_n is at least 1 + log r/(100 r); a claimed resolution of the mean-normalized de Bruijn–Erdős conjecture, unrefereed and unreviewed.

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Samuel Korsky, A resolution of the de Bruijn--Erdős consecutive-gap problem, arXiv:2609.07196, math.CO; v1 of 7 September 2026 (the ratio part only, per the arXiv comment on v2), v2 of 9 September 2026, 16 pages, manuscript dated September 8, 2026. Unrefereed; no journal reference on arXiv. Suggested key [Ko26b].

Retained artifact. The folder-name PDF is arXiv v2, retrieved from https://arxiv.org/pdf/2609.07196v2; 396,636 bytes; the arXiv v2 LaTeX source and HTML rendering were fetched too, and the arXiv metadata sits in the .arxiv/ sidecar. The text layer was read. Version 1 is not held. The external link of the proof claim registered for Problem 1221 on erdosproblems.com (submitted 2026-09-08) is a Google Drive copy of the same manuscript (16 pages, dated September 8, 2026; 436,470 bytes), whose extracted text agrees with the arXiv v2 text apart from the arXiv stamp line (compared here); it is not held. The arXiv record (https://arxiv.org/abs/2609.07196, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Read status. Claims checked for Theorem 1.1 (p. 2) and the consequences stated under it, read clause by clause in the PDF; the proof (Sections 2--8, pp. 4--15) was read for its structure only and nothing in it was checked. No independent review of the argument exists in the sources searched, and the paper is unrefereed. The acknowledgments (p. 15) state that an AI system was used for the literature search that located Larcher's quantitative discrepancy bound, for completing the mathematical argument from the author's two main ideas, for developing and auditing the $L^1$ transport and localization argument, for reviewing the proof and for revising the exposition, and that "The author independently checked the arguments and calculations and assumes full responsibility for all mathematical claims"; the system is not named here. The site's proof-claim note by the author says the document was posted in a preliminary form ahead of the preprint. Standing in this corpus: claimed, unrefereed, unreviewed.

Overview

The paper takes a sequence $(x_n)_{n\ge1}$ of distinct points on $\mathbb T=\mathbb R/\mathbb Z$, the $n$ gaps cut by the first $n$ points, and the $r$-spans, the sums of $r$ consecutive gaps; $M_n^{(r)}$ and $m_n^{(r)}$ are the largest and smallest $r$-spans, the mean $r$-span is $r/n$, and

$$\bar A_r=\inf_X\limsup_{n\to\infty}nM_n^{(r)},\qquad \underline A_r=\sup_X\liminf_{n\to\infty}nm_n^{(r)},\qquad \mu_r=\inf_X\limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}},$$

over sequences $X$ of distinct points (p. 1--2). These are the constants $\Lambda_r$, $\lambda_r$, $\mu_r$ of de Bruijn and Erdős 1949 and of the site's Problem 1221, restricted to sequences of distinct points. The paper restates the 1949 conjecture as $\bar A_r-r\to\infty$, $r-\underline A_r\to\infty$ and $r(\mu_r-1)\to\infty$ (p. 2), which is the mean-normalized reading of the note's Section 6 wording, and notes (p. 3) that in the notation $\hat M=M/r$, $\hat m=m/r$ its first two quantities are exactly $\limsup r(n\hat M_n^{(r)}-1)$ and $\limsup r(1-n\hat m_n^{(r)})$.

Theorem 1.1 (p. 2) asserts that for some absolute constants $c>0$ and $r_0$, each integer $r\ge r_0$ and each sequence of distinct points satisfy $\limsup_n(nM_n^{(r)}-r)\ge c\sqrt{\log r}$, $\limsup_n(r-nm_n^{(r)})\ge c\sqrt{\log r}$ and $\limsup_nM_n^{(r)}/m_n^{(r)}\ge1+\log r/(100r)$; hence $\bar A_r-r\ge c\sqrt{\log r}$, $r-\underline A_r\ge c\sqrt{\log r}$ and $\mu_r-1\ge\log r/(100r)$ for all large $r$. With the upper bound $\mu_r\le1+C\log r/r$ of Clément and Steinerberger this would fix the order of $\mu_r-1$ at $\log r/r$ and answer a question of Brethouwer (Ph.D. thesis, TU Delft 2024, Section 3.3.1, Question 3, as the paper cites it; not read here). The paper cites the author's fixed-$r$ bound $\mu_r\ge1+r/(r^2-1)$ for $r\ge2$ (the 2026 note), an unpublished entropy-potential manuscript of the author (not used), Bevan's small-$r$ constructions (arXiv:2607.00775) and the finite-horizon study of DeLeo, Henderschedt and Wells (arXiv:2605.29166).

The method, as the paper outlines it (p. 3) and as its sections are organized: Section 2 (p. 4) compares interval counts at nearby times by composing forward and backward cyclic moves by $kr$ places, which are bijections of nested point sets, so that a bound on the normalized span error $a_t+b_t\le A$ (display (2.1)) controls the counts $U_t(D)$ and $V_t(D)$ in intervals of length $D/t$ (Lemma 2.1). Section 3 (p. 6) iterates this to short intervals (Proposition 3.1: counting error at most $3A+O(A/\log(r/A))$ on intervals holding about $S=\sqrt{Ar}/\log^2(r/A)$ points). Section 4 (p. 7) states a finite-prefix form of Schmidt's discrepancy theorem, Theorem 4.1, $H_L\ge\frac1{16}\log L$ for every list of $L\ge L_0$ points in $[0,1)$, derived from Larcher's proof (J. Complexity 31 (2015)) rather than quoted from a published statement, and Lemma 4.2 transfers short-interval counting bounds to such a list. Section 5 (p. 9) proves the ratio assertion by contradiction, taking $A=(\log r)/100$ and comparing $\frac3{100}\log r$ with $\frac1{32}\log r$; Remark 5.1 says the constant $1/100$ is not optimized. Sections 6--8 (pp. 10--15) prove the two one-sided assertions: under either one-sided hypothesis (6.1) the mean-span identity gives $L^1$ control of all spans (Lemma 6.1), the averaged walk comparison gives $L^1$ control of short-interval counts (Lemmas 6.2--6.3, Proposition 6.4), and localizing the points of a moving short interval with insertion time as a second coordinate lets Halász's planar $L^1$ discrepancy theorem (Theorem 7.1, from Recent Progress in Analytic Number Theory, vol. 2, 1981) force an error of order $\sqrt{\log L}$ (Lemma 7.2), which Section 8 (p. 15) turns into a contradiction for large $r$.

Relation to Problem 1221

Theorem 1.1 addresses the mean-normalized reading of the three-part question, which is the only nontrivial reading of its first two parts (the site's literal $r(\Lambda_r-1)$ is trivially unbounded and its literal $r(1-\lambda_r)$ tends to $-\infty$; see the conjecture page). Its third part is the site's third part restricted to sequences of distinct points. Two fidelity points are recorded on the result page: the theorem's infimum and supremum run over sequences of distinct points, while neither the site's wording nor the 1949 note's Section 1 excludes coincident points, and whether the constants agree over the two families is not settled in the sources read; and the paper's constants $c$, $r_0$ are unspecified. The site shows the problem OPEN with this as its one registered proof claim, the community database keeps the entry open, and the two site comments on the claim (10 September 2026) discuss the speed of posting and the attribution of the AI's role, not the mathematics. The preprint therefore does not change the problem's status here.

Bears on. Problem 1221: a claimed proof of all three parts under the mean-normalized reading, for distinct points; claimed, unrefereed, unreviewed.