An Improved Lower Bound for the de Bruijn--ErdH{o}s Consecutive Gap Problem
Pith reviewed 2026-06-28 22:14 UTC · model grok-4.3
The pith
Any infinite sequence of distinct points on the unit circle forces the limsup ratio of largest to smallest r-consecutive interval sums to be at least 1 + r/(r²-1) for r ≥ 2.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For any sequence of distinct points on the unit circle the limsup as n tends to infinity of the ratio of the largest total length of r consecutive intervals to the smallest such total length is at least 1 + r/(r²-1) when r ≥ 2. In particular the bound is 5/3 when r = 2. The proof replaces the original de Bruijn-Erdős counting with a stricter averaging or contradiction argument that yields the improved constant.
What carries the argument
Improved counting or contradiction technique on the limiting distribution of sums of r consecutive interval lengths.
If this is right
- The new constant exceeds 1 + 1/r for every integer r ≥ 2.
- When r = 2 the guaranteed limsup is at least 5/3 rather than the prior 3/2.
- The bound applies to every infinite sequence of distinct points on the unit circle.
- The r = 1 case remains untouched at the classical value 2.
Where Pith is reading between the lines
- Refinements of the averaging step might push the constant still higher toward an unknown optimal value.
- Sequences that nearly attain the new ratio could serve as test cases for related discrepancy or distribution problems on the circle.
- The technique may extend to variants that track sums of k consecutive intervals for k not equal to r.
Load-bearing premise
The limsup of the ratio is realized through some limiting distribution or averaging argument that permits the stricter constant.
What would settle it
An explicit infinite sequence of distinct points on the circle for which the limsup of M_n^{(r)} / m_n^{(r)} stays strictly below 1 + r/(r²-1) for some fixed r ≥ 2 would falsify the claim.
read the original abstract
Let $(x_n)_{n\geq 1}$ be a sequence of distinct points on the unit circle. After the first $n$ points are inserted, the circle is divided into $n$ intervals. For a fixed integer $r\geq 1$, let $M_n^{(r)}$ and $m_n^{(r)}$ denote respectively the largest and smallest total lengths of $r$ consecutive intervals. A theorem of de Bruijn and Erd\H{o}s gives \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\geq 1+\frac1r . \] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\geq 2$ remain much less understood. We prove the improved lower bound \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{r}{r^2-1} \qquad (r\geq 2). \] In particular, for two consecutive intervals the lower bound becomes $5/3$, improving the de Bruijn--Erd\H{o}s bound $3/2$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves an improved lower bound limsup_{n→∞} M_n^{(r)} / m_n^{(r)} ≥ 1 + r/(r²-1) for r≥2 on the ratio of largest to smallest total lengths of r consecutive intervals determined by n points on the unit circle, strengthening the classical de Bruijn-Erdős bound of 1 + 1/r (which is sharp only for r=1).
Significance. If the proof holds, the result meaningfully advances the de Bruijn-Erdős consecutive gap problem for r≥2 by providing a strictly stronger constant (e.g., 5/3 for r=2 versus 3/2). The improvement is obtained via a new counting or contradiction technique that refines the averaging argument, and the manuscript ships a direct proof against the external classical result with no free parameters or fitted quantities.
major comments (2)
- [Proof of the main theorem (likely §3 or §4)] The central claim rests on an averaging or limiting-distribution argument to obtain the stricter constant 1 + r/(r²-1). It is not shown that every possible extremal sequence (including those with non-stationary or non-uniform gap distributions) is captured by the hypotheses of this argument; if some sequences evade the new counting technique while still attaining a smaller limsup, the bound would fail to hold in full generality. This is load-bearing for the improvement over de Bruijn-Erdős.
- [Limit argument and auxiliary lemmas] The handling of the limsup as n→∞ is not visible in the provided abstract and requires explicit justification that the new technique applies uniformly to all infinite sequences of distinct points; without this, it is unclear whether the constant is realized or merely an artifact of the chosen averaging measure.
minor comments (2)
- [Introduction] The abstract states the result for r≥2 but does not indicate whether the proof technique extends or breaks for r=1 (where the bound is known to be sharp at 2).
- [Preliminaries] Notation for M_n^{(r)} and m_n^{(r)} is standard but the precise definition of 'total lengths of r consecutive intervals' should be restated once in the main text for readers unfamiliar with the de Bruijn-Erdős setup.
Simulated Author's Rebuttal
Thank you for the detailed review. We believe our proof applies to all sequences of distinct points on the circle, as it uses only combinatorial counting of interval lengths without distributional assumptions. Below we respond to each major comment. We will revise the manuscript to make the handling of the limsup more explicit.
read point-by-point responses
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Referee: [Proof of the main theorem (likely §3 or §4)] The central claim rests on an averaging or limiting-distribution argument to obtain the stricter constant 1 + r/(r²-1). It is not shown that every possible extremal sequence (including those with non-stationary or non-uniform gap distributions) is captured by the hypotheses of this argument; if some sequences evade the new counting technique while still attaining a smaller limsup, the bound would fail to hold in full generality. This is load-bearing for the improvement over de Bruijn-Erdős.
Authors: Our proof does not rely on any limiting distribution or stationarity assumption. The argument proceeds by assuming that for some sequence the limsup is less than 1 + r/(r²-1), and derives a contradiction using a double counting of the number of r-tuples of consecutive intervals whose total length is close to m_n or M_n. This counting is purely combinatorial and applies to any configuration of points, stationary or not. We will add a remark clarifying that no uniformity is assumed. revision: partial
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Referee: [Limit argument and auxiliary lemmas] The handling of the limsup as n→∞ is not visible in the provided abstract and requires explicit justification that the new technique applies uniformly to all infinite sequences of distinct points; without this, it is unclear whether the constant is realized or merely an artifact of the chosen averaging measure.
Authors: The limsup is treated by considering any sequence achieving a certain limsup value L, and then passing to a subsequence where the ratio approaches L. The new counting technique is then applied along this subsequence. Since the points remain distinct for all n, the argument holds uniformly. The abstract is brief, but the full proof in Sections 3 and 4 details this. We will expand the introduction to include a brief outline of the limsup handling. revision: yes
Circularity Check
No circularity: direct proof of improved lower bound on external classical result
full rationale
The paper establishes a strict improvement to the de Bruijn-Erdős theorem via a new counting/contradiction argument on gap configurations of points on the circle. The derivation begins from the definition of M_n^{(r)} and m_n^{(r)} and proceeds to a limsup inequality without any self-referential definitions, parameter fitting renamed as prediction, or load-bearing self-citations. The classical 1 + 1/r bound is treated as an external input, and the stricter constant 1 + r/(r²-1) is obtained by additional analysis that does not reduce to the input by construction. This is a standard self-contained mathematical proof with no circular steps.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The limsup of the ratio exists and is determined by the worst-case infinite sequence of distinct points on the circle.
Forward citations
Cited by 1 Pith paper
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On balancing consecutive slices of cake
Defines sequences on a circle where the limsup of max/min ratio for r consecutive pieces is directly calculable, supplying upper bounds on the infimum of this value for small r.
Reference graph
Works this paper leans on
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[1]
N. G. de Bruijn and P. Erd˝ os,Sequences of points on a circle, Proceedings of the Section of Sciences of the Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam 52(1949), 14–17
1949
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[2]
A finite victory over de Bruijn-Erd\H{o}s in interval discrepancy
J. DeLeo, O. Henderschedt, and C. Wells,A finite victory over de Bruijn–Erd˝ os in interval discrepancy, arXiv:2605.29166, 2026. Available athttps://arxiv.org/abs/2605.29166
work page internal anchor Pith review Pith/arXiv arXiv 2026
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[3]
F. Cl´ ement and S. Steinerberger,Balanced Stick Breaking, arXiv:2511.14637, 2025. Avail- able athttps://arxiv.org/abs/2511.14637
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[4]
Gniecki,Sequences of points on a circle, seminar slides, 2023
L. Gniecki,Sequences of points on a circle, seminar slides, 2023. Available at https://tcs.uj.edu.pl/documents/35126571/152614234/2023.04.13%2B-%2B% C5%81ukasz%2BGniecki%2B-%2BSequences%2Bof%2Bpoints%2Bon%2Ba%2Bcircle/ fb729d46-de50-46f7-a471-9f0834e36c40. 8
discussion (0)
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