Problem 1221

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claims/: The 1 claim page of Problem 1221, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a=(a_1,\ldots,a_n)$ be points, ordered consecutively, around the circle (normalised to have circumference $1$). Let

$$M_r(a)=\max_r \lvert a_{i+r}-a_i\rvert$$

and

$$m_r(a)=\min_r \lvert a_{i+r}-a_i\rvert,$$

where the indices are interpreted in a cyclic fashion and the distance is around the circle (in other words, the maximum and minimum distance between $r$-consecutive points). Let

$$\Lambda_r=\inf_a \limsup_n n M_r(a_1,\ldots,a_n),$$ $$\lambda_r = \sup_a \liminf_n nm_r(a_1,\ldots,a_n),$$

and

$$\mu_r=\inf_a \limsup_n \frac{M_r(a)}{m_r(a)},$$

where the outer $\inf/\sup$ is over all infinite sequences $a=(a_1,\ldots)$. Is it true that $r(\Lambda_r-1)$, $r(1-\lambda_r)$, and $r(\mu_r-1)$ tend to infinity as $r\to \infty$?

Statement (corrected). Let $a=(a_1,\ldots,a_n)$ be points, ordered consecutively, around the circle (normalised to have circumference $1$). Let

$$M_r(a)=\max_r \lvert a_{i+r}-a_i\rvert$$

and

$$m_r(a)=\min_r \lvert a_{i+r}-a_i\rvert,$$

where the indices are interpreted in a cyclic fashion and the distance is around the circle (in other words, the maximum and minimum distance between $r$-consecutive points). Let

$$\Lambda_r=\inf_a \limsup_n n M_r(a_1,\ldots,a_n),$$ $$\lambda_r = \sup_a \liminf_n nm_r(a_1,\ldots,a_n),$$

and

$$\mu_r=\inf_a \limsup_n \frac{M_r(a)}{m_r(a)},$$

where the outer $\inf/\sup$ is over all infinite sequences $a=(a_1,\ldots)$. Is it true that $\Lambda_r-r$, $r-\lambda_r$, and $r(\mu_r-1)$ tend to infinity as $r\to \infty$?

Notes. With the site's definitions, $M_r$ and $m_r$ are the largest and smallest sums of $r$ consecutive gaps (the "distance between $r$-consecutive points" is the forward arc through the $r-1$ intervening points), whose mean is $r/n$, so $nM_r\ge r\ge nm_r$. The site's first two expressions then fail as a question. Section 3 of [dBEr49] gives $\Lambda_r\ge1/\log(1+1/r)>r$, so $r(\Lambda_r-1)\ge r(r-1)$ and the first tends to infinity trivially. A sum of $r$ consecutive gaps is at least $r$ times the smallest gap, so $\lambda_r\ge r\lambda_1=r/\log4>1$ for $r\ge2$, and $r(1-\lambda_r)\le r(1-r/\log4)$ is already negative at $r=2$ and tends to $-\infty$; read as printed, the question has the answer no. The library's conjecture page writes the argument out. The change replaces "$r(\Lambda_r-1)$, $r(1-\lambda_r)$" by "$\Lambda_r-r$, $r-\lambda_r$", which normalizes $\Lambda_r$ and $\lambda_r$ by the mean span: $r(\Lambda_r/r-1)=\Lambda_r-r$ and $r(1-\lambda_r/r)=r-\lambda_r$. The third expression needs no normalization and is unchanged. The evidence is the posers' own words in [dBEr49]. The introduction (p. 14) says: "All we can prove is that $\mu_r\geq 1+1/r$ (and analogous inequalities for $\Lambda_r$ and $\lambda_r$); we conjecture that $r(\mu_r-1)$ is unbounded." Section 6 (p. 17) opens: "The inequalities (3.3), (4.3) and (5.7) are probably not best possible if $r\geq2$." It then conjectures that the three expressions tend to infinity. Normalized by $r$, the three bounds are analogous, of the form $1\pm c/r$, and give exactly the bounded lower bounds $\Lambda_r-r\ge\tfrac12-\tfrac1{12r}+O(r^{-2})$, $r-\lambda_r\ge\tfrac12-\tfrac5{12r}+O(r^{-2})$ and $r(\mu_r-1)\ge1$ that the conjecture says can be improved to growth. With the printed normalization, (3.3) would already prove the first part, and the second part would be false, so the posers' words are true only of the normalized form. The form is the one Korsky [Ko26b, p. 2] prints when he restates the conjecture of [dBEr49, p. 17], $\bar A_r-r\to\infty$, $r-\underline A_r\to\infty$, $r(\mu_r-1)\to\infty$, apart from the hypotheses of his theorem (his constants are taken over sequences of distinct points, a restriction the correction does not adopt; see Formulation). The form is taken from these two sources, not from the results that bear on it. The defect is already in the source: Section 6 of [dBEr49] prints $r(\Lambda_r-1)$ and $r(1-\lambda_r)$ with the same definitions, and the site reproduces it; the site's page shows the community database's note that the original source is ambiguous. The failure was reported by the GitHub user OutBlade as issue 414 of the community database (teorth/erdosproblems#414, 11 September 2026), with a note at OutBlade/erdos-notes that proposes the missing factor of $r$; both say the observation was found and checked with Claude (Anthropic). The database's pull request 416, merged 18 September 2026, marks the entry "ambiguous statement". This is the one result about the site's wording; it is credited here and counts for nothing. Unread: the Brethouwer thesis (TU Delft 2024), which [Ko26b] and [ClSt25] cite for the third part.

Formulation. The Statement is the site's wording as accessed 2026-09-27 (page last edited 7 September 2026; its one revision, of 2026-09-07 07:09:11, corrected a typo in the definitions of $M_r$ and $m_r$). It reproduces Section 6 of [dBEr49] (printed p. 17; the site's "p. 6" is the offprint page), with the normalization slip that the Notes record. [dBEr49] gives $\Lambda_r\ge1/\log(1+1/r)>r$ (Section 3) and $\lambda_r\le\frac{r}{r+1}/\log(1+1/r)<r$ (4.3); in the corrected form these bound $\Lambda_r-r$ and $r-\lambda_r$ below by $\tfrac12+o(1)$, (5.7) bounds $r(\mu_r-1)$ below by $1$, and the question is whether each tends to infinity. The third expression is the same in both forms. One further scope point: neither the site's wording nor Section 1 of [dBEr49] (numbers mod $1$; $n$ intervals of total length $1$) excludes coincident points, while the 2026 preprint [Ko26b] takes its infimum and supremum over sequences of distinct points; whether the constants over the two families agree is not settled in the sources.

Status. The site's label is OPEN. A preprint [Ko26b] (arXiv:2609.07196v2, 9 September 2026) claims all three parts of the corrected Statement for sequences of distinct points, and is registered on the site as a full proof claim (submitted 2026-09-08); it is recorded as the pending partial claim Korsky 2026, since its constants run over sequences of distinct points only, and the frontmatter standing is open. No acceptance evidence was found in a search: the preprint is unrefereed, the site shows OPEN, the community database keeps open (2026-09-18), and no independent review exists.

Source. erdosproblems.com/1221, accessed 2026-09-27: the problem page (OPEN, with the site's note that no finite computation can settle it; last edited 7 September 2026; source key [dBEr49, p. 6]; no formalized statement; the note that the original source is ambiguous), its LaTeX source, its empty discussion thread, its proof-claims tab (one full proof claim of 2026-09-08 with two comments of 10 September 2026) and its history (one revision). Cite as: T. F. Bloom, Erdős Problem #1221, https://www.erdosproblems.com/1221, accessed 2026-09-27.

References.

• [dBEr49] de Bruijn, N. G. and Erdős, P., Sequences of points on a circle. Nederl. Akad. Wetensch., Proc. 52 (1949), 14--17 = Indagationes Math. 11 (1949), 46--49 (communicated 18 December 1948; MR 33331). Definitions and the $r=1$ values, Sections 1--2, pp. 14--15; the lower bound for $\Lambda_r$, Section 3, p. 15 (unnumbered on the page, cited as (3.3) in Section 6; footnote 2 on p. 15 credits the Section 4 proof as found independently by van Aardenne-Ehrenfest); the upper bound (4.3) for $\lambda_r$, p. 16; (5.1) and (5.7) for $\mu_r$, pp. 16--17; the conjecture, Section 6, p. 17. The publisher's version of record is on the TU/e research portal (a cover sheet and four image-only article pages). Library home: debruijn_erdos_1949_sequences_points_circle.
• [Ko26b] Korsky, S., A resolution of the de Bruijn–Erdős consecutive-gap problem. arXiv:2609.07196 (v1 7 September 2026, the ratio part only; v2 9 September 2026, all three parts; 16 pages; math.CO). Theorem 1.1, p. 2. The site's proof-claim link is the same manuscript (dated September 8, 2026) on Google Drive. Unrefereed; the paper's acknowledgments (p. 15) state that GPT Astra was used to locate Larcher's bound and to complete, audit and review the argument, with the author checking it independently. Library home: korsky_2026_resolution_debruijn_erdos_consecutive_gap_problem.
• [Ko26a] Korsky, S., An improved lower bound for the de Bruijn–Erdős consecutive gap problem. arXiv:2605.30959v1 (29 May 2026; 8 pages). Theorem 1.1, p. 2: $\mu_r\ge1+r/(r^2-1)$ for $r\ge2$. Unrefereed. Library home: korsky_2026_improved_lower_bound_debruijn_erdos_consecutive_gap_problem.
• [ClSt25] Clément, F. and Steinerberger, S., Balanced stick breaking. arXiv:2511.14637v1 (18 November 2025; 12 pages). Theorem 2, p. 2: $\mu_r\le1+c\log r/r$ for every $r\ge2$. Unrefereed. Library home: clement_steinerberger_2025_balanced_stick_breaking.
• [Be26] Bevan, D., On balancing consecutive slices of cake. arXiv:2607.00775v2 (7 July 2026; 8 pages). Upper bounds on $\inf_a\mu_r(a)$ for small $r$ from a family of sequences whose $\mu_r(a)$ is computable. Known from its abstract; lead.
• Leads: DeLeo, J., Henderschedt, O. and Wells, C., A finite victory over de Bruijn–Erdős in interval discrepancy, arXiv:2605.29166 (the finite $r=1$ splitting problem); Brethouwer, J.-T. F., Ph.D. thesis, TU Delft 2024, doi 10.4233/uuid:617757b7-32d2-4bac-8be3-5a6a13e6e710, Section 3.3.1, Question 3 (whether $\mu_r-1\gg\log r/r$), as cited by [Ko26b] and [ClSt25]; van Aardenne-Ehrenfest, Proc. 48 (1945), 266--271 = Indag. Math. 7 (1946), 71--76, the "just distributions" theorem that [dBEr49] says would follow from the third part; a second Bevan preprint on two-slice portions, listed by Semantic Scholar as citing [Ko26a] and not located on arXiv by title.

Formalization. None recorded. The site shows "Formalised statement? No"; formal-conjectures at main (directory listed 2026-09-27) has no 1221.lean; conjectures.io lists no problem or result for 1221 (results and problems pages,).

Current assessment

The question (site formulation of 2026-09-27). The statement above, OPEN, last edited 7 September 2026, source key [dBEr49, p. 6], with the site's note that the original source is ambiguous. The commentary quotes the $r=1$ values $\Lambda_1=1/\log2$, $\lambda_1=1/\log4$, $\mu_1=2$ with the witness $a_k=\log_2(2k-1)$, and the three bounds $\Lambda_r\ge1/\log(1+1/r)$, $\lambda_r\le\frac r{r+1}/\log(1+1/r)$, $\mu_r\ge1+1/r$. The site's first two expressions lack the normalization by $r$, as the Notes record: as printed, the first is trivially unbounded and the second tends to $-\infty$. The corrected Statement asks whether $\Lambda_r-r$, $r-\lambda_r$ and $r(\mu_r-1)$ all tend to infinity, the form [Ko26b] restates (as $\bar A_r-r\to\infty$, $r-\underline A_r\to\infty$, $r(\mu_r-1)\to\infty$, a corrected paraphrase rather than the note's wording) and issue 414 of the community database proposes (a missing factor of $r$).

Supported status and best progress. For the corrected Statement nothing refereed goes beyond [dBEr49]: $\Lambda_r-r\ge\tfrac12+o(1)$ (Section 3), $r-\lambda_r\ge\tfrac12+o(1)$ ((4.3)), $\mu_r\ge1+1/r$ ((5.7)), and the exact $r=1$ values (Section 2). Unrefereed progress: $\mu_r\ge1+r/(r^2-1)$ for $r\ge2$ over sequences of distinct points (Theorem 1.1 of Ko26a); $\mu_r\le1+c\log r/r$ for every $r\ge2$ (Theorem 2 of ClSt25), so the third part can grow at most like $\log r/r$; small-$r$ upper bounds on $\inf_a\mu_r(a)$ in [Be26] (abstract only).

The claim. Theorem 1.1 of Ko26b claims absolute constants $c>0$ and $r_0$ such that for $r\ge r_0$ and every sequence of distinct points on the circle, $\limsup_n(nM_n^{(r)}-r)\ge c\sqrt{\log r}$, $\limsup_n(r-nm_n^{(r)})\ge c\sqrt{\log r}$ and $\limsup_nM_n^{(r)}/m_n^{(r)}\ge1+\log r/(100r)$, hence all three parts of the corrected Statement for sequences of distinct points, with $\mu_r-1$ of exact order $\log r/r$ together with [ClSt25]. Method as the paper outlines it: comparison of interval counts at nearby insertion times through forward and backward cyclic walks; a finite-prefix form of Schmidt's discrepancy theorem derived from Larcher's proof for the ratio; Halász's planar $L^1$ discrepancy theorem for the two one-sided bounds. Version 1 (7 September 2026) proved the ratio part only; version 2 added the two one-sided parts. It is registered on the site as a full proof claim (2026-09-08), with the author's note that the document was posted in a preliminary form ahead of the preprint; the paper's acknowledgments (p. 15) state that GPT Astra was used to complete, audit and review the argument and that the author independently checked it, the site's proof-claim tab registers the claim as made using GPT Astra, and in the thread the author attributes the two one-sided bounds to the system, built on his ratio work. The two site comments (10 September 2026) concern the speed of posting and the attribution of the AI's role, not the mathematics. Acceptance: none found (unrefereed; no journal reference; Crossref finds no published version; site OPEN; community database open on 2026-09-18; no independent review; no citing evaluation located). The claim's scope is the distinct-point family (Formulation). The claim page Korsky 2026 records it as a pending partial claim, which leaves the standing open.

Reconstructions. The proofs of the three 1949 bounds, of the fixed-$r$ improvement [Ko26a], of the upper bound [ClSt25] and of the claimed resolution [Ko26b], each with the reading of the ambiguous statement it addresses and with its imported inputs labeled, are reconstructed (author-recorded, unreviewed; no status or tier follows) in the Problem 1221 research folder.

Search scope (2026-09-27). erdosproblems.com (page, LaTeX source, history, discussion thread, proof-claims tab and its comments, the bibliography entry dBEr49, the reference's problem list: 1220 and 1221); the community database teorth/erdosproblems (the YAML copy and live main, entry 1221 open, last update 2026-09-12, comment "ambiguous statement", the highest number in the database; issue 414; pull request 416, "Add problems 1218--1221; mark 1221 as ambiguous", merged 2026-09-18); conjectures.io (results and problems pages); formal-conjectures main (directory listing; issue and pull-request search); arXiv (API queries for Korsky and circle, and for de Bruijn, Erdős and circle; the abstract pages of 2609.07196 v1 and v2, 2605.30959, 2511.14637, 2607.00775 and 2605.29166, none with a journal reference); Crossref (no published version of the three preprints); Semantic Scholar (the record of 2605.30959 with three citing titles; queries for 2609.07196 and 2511.14637 rate-limited); the TU/e list of de Bruijn's publications (the 1949 PDF); the DOI resolver for the Brethouwer thesis. The site's Google Drive proof document is the same manuscript as arXiv v2 of 2609.07196 (text identical apart from the arXiv stamp). Not searched: MathSciNet, zbMATH, Google Scholar, X. Outside the search: the full texts of 2607.00775 and 2605.29166 (abstracts only), the Brethouwer thesis, van Aardenne-Ehrenfest 1945/46, the author's earlier potential-function manuscript on Google Drive.

Remaining gaps. (1) No independent review of [Ko26b]; the ratio part (Sections 2--5, the shorter of the two arguments) is the first candidate for review. (2) For [Ko26a] and [ClSt25] only the main theorems are checked, as statements; [Be26] and the other leads are known from their abstracts. (3) The normalization defect is inherited from [dBEr49], as the Notes record. (4) The claim page would move to accepted only on refereed publication of [Ko26b] or documented independent acceptance (a site status change with review, or a reviewed reproof). (5) [Ko26b] proves its bounds for sequences of distinct points, and whether the constants over all sequences equal those over distinct sequences is not settled in the sources. (6) X was not searched for research announcements.

Status check (2026-10-05). On 2026-10-05 the site showed OPEN (last edited 7 September 2026) with the one full proof claim of 2026-09-08 and its two comments of 10 September 2026; arXiv showed v2 of 9 September 2026 with no journal reference, and no citing paper was found; the community database showed "ambiguous statement" (pull request 416, merged 2026-09-18); Palomar and conjectures.io had no entry; and no acceptance, refereeing or independent human review was found. Two items of context, neither a review: an automated, AI-only review posted on a public review site (10 September 2026, no human sign-off) recommends human refereeing and flags the ratio part as fragile, its final comparison of $(3/100)\log r$ against $(1/32)\log r$ leaving a margin of about $0.00125\log r$ against $O(\log\log r)$ error terms, with Larcher's discrepancy constant taken from the literature without a self-contained check; and an aggregator of AI-assisted results lists the problem as proved, candidate, review pending, unreviewed (9 September 2026). The second Bevan preprint listed among the leads above is arXiv:2607.05330 (6--7 July 2026): a two-slice ($r=2$) ratio of at most $1.755$ for distinct points, fixed $r$ only, known from its abstract.

Known Results

• [dBEr49] Sections 1--2: $\Lambda_1=1/\log2$, $\lambda_1=1/\log4$, $\mu_1=2$, all attained by $a_k=\log_2(2k-1)$ reduced mod $1$; the $r=1$ case is exactly determined (result page).
• [dBEr49] Section 3, final display ((3.3) in Section 6): $\Lambda_r(a)\ge1/\log(1+1/r)>r$ for every sequence and every $r\ge1$; in mean-normalized form $\Lambda_r-r\ge\tfrac12-\tfrac1{12r}+O(r^{-2})$ (result page).
• [dBEr49] (4.3): $\lambda_r(a)\le\frac r{r+1}/\log(1+1/r)<r$ for every sequence; in mean-normalized form $r-\lambda_r\ge\tfrac12-\tfrac5{12r}+O(r^{-2})$ (result page).
• [dBEr49] (5.1) and (5.7): $M_n^r(a)/m_{n+1}^r(a)\ge1+1/r$ for all $r\ge1$, $n\ge1$, hence $\mu_r\ge1+1/r$, sharp for $r=1$ (result page).
• [dBEr49] Section 6, with the p. 14 remark: the conjecture as posed, that $r(\Lambda_r-1)$, $r(1-\lambda_r)$ and $r(\mu_r-1)$ tend to infinity, and that the unboundedness of $r(\mu_r-1)$ would imply van Aardenne-Ehrenfest's "just distributions" theorem; the normalization defect and the mean-normalized reading are recorded there (result page).
• [ClSt25] Theorem 2 (unrefereed; claims checked): $\mu_r\le1+c\log r/r$ for every $r\ge2$, with the golden-ratio Kronecker and base-2 van der Corput sequences as examples; an upper bound on the third part's growth (result page).
• [Ko26a] Theorem 1.1 (unrefereed; claims checked): $\mu_r\ge1+r/(r^2-1)$ for $r\ge2$ over sequences of distinct points, so $\mu_2\ge5/3$; a fixed-$r$ improvement of (5.7) (result page).
• [Ko26b] Theorem 1.1 (claimed; unrefereed, unreviewed, AI-assisted by the paper's own statement): for $r\ge r_0$ and every sequence of distinct points, $\limsup_n(nM_n^{(r)}-r)\ge c\sqrt{\log r}$, $\limsup_n(r-nm_n^{(r)})\ge c\sqrt{\log r}$ and $\limsup_nM_n^{(r)}/m_n^{(r)}\ge1+\log r/(100r)$; a claimed proof of all three parts of the corrected Statement for distinct points, registered on the site as a full proof claim (2026-09-08), with no acceptance evidence (result page; claim page).
• [Be26] (abstract only; lead): explicit upper bounds on $\inf_a\mu_r(a)$ for small $r$ from a family of sequences whose $\mu_r(a)$ is computable.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• clement_steinerberger_2025_balanced_stick_breaking
• clement_steinerberger_2025_balanced_stick_breaking / theorem_2
• debruijn_erdos_1949_sequences_points_circle
• debruijn_erdos_1949_sequences_points_circle / conjecture_p17
• debruijn_erdos_1949_sequences_points_circle / inequality_3_3
• debruijn_erdos_1949_sequences_points_circle / inequality_4_3
• debruijn_erdos_1949_sequences_points_circle / inequality_5_7
• debruijn_erdos_1949_sequences_points_circle / section_2_r_equals_1
• korsky_2026_improved_lower_bound_debruijn_erdos_consecutive_gap_problem
• korsky_2026_improved_lower_bound_debruijn_erdos_consecutive_gap_problem / theorem_1_1
• korsky_2026_resolution_debruijn_erdos_consecutive_gap_problem
• korsky_2026_resolution_debruijn_erdos_consecutive_gap_problem / theorem_1_1