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Source. S. Korsky, A resolution of the de Bruijn--Erdős consecutive-gap problem, arXiv:2609.07196v2 (9 September 2026), Theorem 1.1 (p. 2), Section 5 (p. 9, the ratio assertion, with Remark 5.1) and Section 8 (p. 15, the one-sided assertions) of the retained PDF, read in the canonical conversion beside the PDF and checked against the text layer; held by its library card, Korsky 2026, resolution, with the result page Theorem 1.1. The inputs are reconstructed on the pages for Lemma 2.1, Proposition 3.1, Lemma 4.2 with Theorem 4.1, Lemma 6.1, Lemma 6.2, Lemma 6.3, Proposition 6.4 and Lemma 7.2 with Theorem 7.1.

Standing. Author-recorded reconstruction; not an independent review; changes no status and assigns no tier. The source is an unrefereed, AI-assisted preprint (the card records the paper's own statement of the assistance) registered on erdosproblems.com as a full proof claim for Problem 1221, with no acceptance evidence found on 2026-09-27; the problem page keeps status open. Two inputs are imported and not checked here: the finite-prefix discrepancy bound (Theorem 4.1, derived by the source from Larcher's proof) and Halász's planar $L^1$ discrepancy theorem (Theorem 7.1). Everything else in the chain is written out on the linked pages.

Definitions

Let $(x_n)_{n\ge1}$ be distinct points of $\mathbb T=\mathbb R/\mathbb Z$. After the first $n$ points are inserted, the $n$ gaps are listed in cyclic order; an $r$-span is the sum of $r$ consecutive gaps, and $M_n^{(r)}$, $m_n^{(r)}$ are the largest and smallest $r$-spans ($n\ge r$). The mean $r$-span is $r/n$, so $nm_n^{(r)}\le r\le nM_n^{(r)}$. Over sequences $X$ of distinct points,

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

Statement (Theorem 1.1, p. 2)

There are absolute constants $c>0$ and $r_0\in\mathbb N$ such that, for every integer $r\ge r_0$ and every sequence of distinct points on $\mathbb T$,

$$\limsup_{n\to\infty}\bigl(nM_n^{(r)}-r\bigr)\ \ge\ c\sqrt{\log r},\qquad \limsup_{n\to\infty}\bigl(r-nm_n^{(r)}\bigr)\ \ge\ c\sqrt{\log r},$$

and

$$\limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\ \ge\ 1+\frac{\log r}{100\,r}.$$

Consequently $\bar A_r-r\ge c\sqrt{\log r}$, $r-\underline A_r\ge c\sqrt{\log r}$ and $\mu_r-1\ge\log r/(100r)$ for all sufficiently large $r$.

Proof of the ratio assertion (Section 5)

Fix a sufficiently large $r$ and suppose, for a contradiction, that

$$\limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\ <\ 1+\frac{\log r}{100r}.$$

Put $A=(\log r)/100$; for $r\ge e^{100}$ this gives $A\ge1$. Since the ratio is at least $1$, there is a $C$ with $0<C<A$ such that $M_n^{(r)}/m_n^{(r)}\le1+C/r$ for all sufficiently large $n$. Suppress the superscript $(r)$.

Pointwise span control. From $nm_n\le r\le nM_n$,

$$n(M_n-m_n)\ \le\ nm_n\cdot\frac Cr\ \le\ C,\qquad nM_n\ \le\ nm_n\Bigl(1+\frac Cr\Bigr)\ \le\ r+C .$$

For real $t$ with $n=\lfloor t\rfloor$ define $a_t=r-nm_n\ge0$ and $b_t=tM_n-r\ge nM_n-r\ge0$. Every $r$-span of $P_t=P_n$ lies between $m_n$ and $M_n$, and

$$\frac{r-a_t}t=\frac nt\,m_n\ \le\ m_n\ \le\ M_n=\frac{r+b_t}t ,$$

while

$$a_t+b_t=n(M_n-m_n)+(t-n)M_n\ \le\ C+\frac{r+C}n\ <\ A$$

for all sufficiently large $t$, because $C<A$ strictly. So hypothesis (2.1) of Lemma 2.1 holds with this $A$.

Short-interval counts. Since $r/A=100r/\log r\to\infty$, the condition $r\ge C_0A$ of Proposition 3.1 holds for large $r$, and (3.1) gives, at all sufficiently large integer times $n$ and for all $x\in\mathbb T$ and $0\le D\le S$,

$$\bigl|N_n((x,x+D/n])-D\bigr|\ \le\ B,\qquad S=\frac{\sqrt{Ar}}{\log^2(r/A)},\qquad B=3A+\frac{C_1A}{\log(r/A)} ,$$

which is hypothesis (4.1) of Lemma 4.2, with $B\ge1$ and $S\ge2$.

Comparison of the two logarithms. With $A=(\log r)/100$, $\log(r/A)=\log r-\log\log r+\log100$, so $C_1A/\log(r/A)=O(1)$ and

$$B=\frac3{100}\log r+O(1).$$

Also $\log S=\frac12\log A+\frac12\log r-2\log\log(r/A)$ with $\log A=\log\log r-\log100$ and $\log\log(r/A)=\log\log r+o(1)$, so

$$\log\lfloor S\rfloor=\frac12\log r-\frac32\log\log r+O(1),$$

with absolute implied constants; in particular $S\to\infty$, so $\lfloor S\rfloor\ge L_0$ for large $r$. Lemma 4.2 now gives

$$\frac3{100}\log r+O(1)\ \ge\ \frac1{16}\log\lfloor S\rfloor =\frac1{32}\log r-O(\log\log r),$$

which is false for all sufficiently large $r$ because $3/100<1/32$. All the constants that govern how large $r$ must be ($e^{100}$, $C_0$, $C_1$, $L_0$ and the implied constants) are absolute, so the threshold $r_0$ is independent of the sequence. This proves the ratio assertion. Remark 5.1 says the coefficient $1/100$ is chosen for simplicity and not optimized; the margin used is $3/100<1/32$.

Proof of the one-sided assertions (Section 8)

Let $C_2,C_3$ be the constants of Proposition 6.4 and $c_4,S_0$ those of Lemma 7.2, and fix an absolute $c>0$ with

$$c\ <\ \frac{c_4}{2C_3\sqrt3}.$$

Suppose first, for a contradiction, that $\limsup_n(nM_n^{(r)}-r)<c\sqrt{\log r}$. Put

$$A=c\sqrt{\log r},\qquad S=\frac{\sqrt{Ar}}{\log^2(r/A)}.$$

For sufficiently large $r$: $A\ge1$; $r\ge C_2A$; the first alternative of hypothesis (6.1), $nM_n^{(r)}-r\le A$, holds for every sufficiently large $n$ (the upper limit is less than $A$); $S\ge S_0$; and $C_3A\ge1$. Proposition 6.4 gives hypothesis (7.1) of Lemma 7.2 with $B=C_3A$ at all late integer times, and Lemma 7.2 gives

$$C_3A\ \ge\ c_4\sqrt{\log S}. \tag{8.1}$$

For this $A$, $\log A=\log c+\frac12\log\log r$ and $\log(r/A)=\log r-\log A$, so

$$\log S=\frac12\log r+O(\log\log r),$$

and in particular $\log S\ge(\log r)/3$ for all sufficiently large $r$. Then (8.1) gives $C_3c\sqrt{\log r}\ge c_4\sqrt{(\log r)/3}$, that is, $c\ge c_4/(C_3\sqrt3)$, contrary to the choice of $c$. This proves the first assertion. If instead $\limsup_n(r-nm_n^{(r)})<c\sqrt{\log r}$, the identical argument runs with the second alternative of (6.1), $r-nm_n^{(r)}\le A$, which is all that Lemmas 6.1--6.3 and Proposition 6.4 use; the same absolute constants serve. This proves the second assertion.

Consequences. For $r\ge r_0$ the three bounds hold for every sequence of distinct points, so $\bar A_r-r\ge c\sqrt{\log r}$, $r-\underline A_r\ge c\sqrt{\log r}$ and $\mu_r-1\ge\log r/(100r)$. With the upper bound $\mu_r\le1+C\log r/r$ of Clément and Steinerberger this places $\mu_r-1$ between two constant multiples of $\log r/r$.

Imported inputs and gaps

• Theorem 4.1 (finite-prefix discrepancy, $H_L\ge\frac1{16}\log L$ for $L\ge L_0$). Stated by the source as a consequence of Section 3 of Larcher's 2015 proof; Larcher's paper is not held and the derivation is not checked. The constant $1/16$ is what makes $1/100$ work. An authored remark on the Lemma 4.2 page notes that the qualitative form $H_L\ge c\log L-1$ follows from Schmidt's planar theorem, which would give the ratio part with an unspecified constant in place of $1/100$.
• Theorem 7.1 (Halász, planar $L^1$ discrepancy). Stated by the source in unnormalized form; the 1981 paper is not held and the statement is not checked against it.
• Distinct points. Lemma 4.2 uses the least distance $\delta>0$ between distinct points of $P_{n_0}$ to keep early points out of the short interval, and all pages use that spans are positive. The 1949 note's constants are defined over sequences that may repeat points; whether they agree with the distinct-point constants is not settled in the sources read.
• Constants. $c$ and $r_0$ are not made explicit. The ratio proof needs $A=(\log r)/100\ge1$, hence $r\ge e^{100}$, before the absolute constants of Proposition 3.1 and Lemma 4.2 enter; the theorem is an asymptotic statement and says nothing for small $r$.
• Nothing else is imported: Lemmas 2.1, 4.2, 6.1--6.3, 7.2 and Propositions 3.1, 6.4 are reconstructed in full on their pages.

Readings addressed

The site's wording of Problem 1221 asks whether $r(\Lambda_r-1)$, $r(1-\lambda_r)$ and $r(\mu_r-1)$ tend to infinity, with $\Lambda_r,\lambda_r,\mu_r$ the 1949 constants over all sequences; the first two expressions are defective as written (the first is at least $r(r-1)$, the second tends to $-\infty$), as the conjecture page records.

• The first two parts of Theorem 1.1 address the mean-normalized reading: with $\hat\Lambda_r=\Lambda_r/r$ and $\hat\lambda_r=\lambda_r/r$ the expressions become $\Lambda_r-r$ and $r-\lambda_r$, and the theorem's $\limsup_n(nM_n^{(r)}-r)$ and $\limsup_n(r-nm_n^{(r)})$ are exactly $\limsup_nr(n\hat M_n^{(r)}-1)$ and $\limsup_nr(1-n\hat m_n^{(r)})$ with $\hat M=M/r$, $\hat m=m/r$ (p. 3 of the source). The 1949 bounds give at least $\frac12+o(1)$ for each (Section 3, (4.3)); the theorem claims $c\sqrt{\log r}$.
• The third part addresses the third expression as written, which needs no normalization; the 1949 bound is $r(\mu_r-1)\ge1$ ((5.7)), the fixed-$r$ improvement is $1+1/(r^2-1)$ (Korsky's note), and the theorem claims $\log r/100$.
• All three parts are stated over sequences of distinct points, a restriction of the site's and the note's family (see above).