On balancing consecutive slices of cake

David Bevan

arxiv: 2607.00775 · v1 · pith:JE4R5QLXnew · submitted 2026-07-01 · 🧮 math.CO

On balancing consecutive slices of cake

David Bevan This is my paper

Pith reviewed 2026-07-02 10:42 UTC · model grok-4.3

classification 🧮 math.CO

keywords circle divisionconsecutive arcsratio of lengthssequence balancingDe Bruijn-Erdős problemlimsup ratioparametric construction

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The pith

A parametric family of point sequences on a circle makes the limsup max-to-min ratio of r consecutive arcs directly computable from construction parameters, yielding upper bounds on the infimum of that ratio over all sequences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines infinite sequences of points on a circle and the ratios that arise when measuring the largest versus smallest of any r consecutive arcs created by the first n points. It introduces a specific family of sequences in which this limsup ratio can be calculated explicitly rather than estimated. The resulting numbers supply concrete upper bounds on the smallest possible such limsup that any sequence can achieve, for small fixed r. A reader would care because the construction turns an existence question about optimal balancing into a finite-parameter optimization problem whose values are immediate to obtain.

Core claim

We define a family of sequences for which the asymptotic least upper bound of this ratio, μ_r(a) = limsup μ^r_n(a), can easily be calculated. Hence, for small r, we present upper bounds on inf μ_r(a).

What carries the argument

A parametric family of sequences on the circle whose limsup ratio μ_r(a) is a direct function of the chosen parameters, serving as an explicit upper bound on the infimum over all sequences.

If this is right

For each small r the quantity inf μ_r(a) is at most the explicit number produced by optimizing the family parameters.
The balancing problem reduces to choosing parameters inside the family rather than searching over all possible sequences.
The same construction supplies a concrete candidate sequence that any purported optimal sequence must at least match in ratio.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same parametric approach might be adapted to produce lower bounds or exact values if the family can be shown to contain a minimizer.
The method could extend to related problems such as balancing arcs under additional constraints like forbidden distances or weighted points.

Load-bearing premise

The constructed family really does achieve a limsup ratio that no other sequence can beat, so the computed value is a genuine upper bound on the infimum.

What would settle it

A different sequence whose computed limsup ratio for some small r falls strictly below the value obtained from the paper's family.

Figures

Figures reproduced from arXiv: 2607.00775 by David Bevan.

**Figure 1.** Figure 1: Cake cutting for recipe σ = 1425367 with two periods. The ratios for portions of five pieces are shown at the right: µ σ 5 = 10/7 is the greatest of these. 3. After bisecting every large piece, we have 2mp equally-sized pieces. We now repeat step 2 with m doubled in value. This repeats ad infinitum, with m successively equal to each power of two. Note that the specific order of the initial p cuts is immate… view at source ↗

**Figure 2.** Figure 2: The leftmost and rightmost positions, with respect to the early periods, of the k(ℓ + 1) + r possibly distinct portions; here k = 3, ℓ = 5 and r = 19. To ensure that all of these can be instantiated, it is sufficient that there are at least ⌈r/ℓ⌉ ⩽ q late periods. Hence m ⩾ 2q suffices to capture all the possible ratios when there are at most q early periods. The creation of more than q early periods does … view at source ↗

**Figure 3.** Figure 3: contains some simple Mathematica code for calculating µ σ r . (This could easily be optimised in a variety of ways.) initLength[r_, p_] := 2 Ceiling[r/p] p splitPiece[a_, k_] := ReplacePart[a, k -> {1, 1}] expandRecipe[s_, len_] := Join @@ (Range[#, len, Length@s]& /@ s) maxMinRatio[r_][a_] := Module[{sizes = Total /@ Partition[Flatten@a, r, 1, {1, 1}]}, Max@sizes / Min@sizes] largestRatio[r_, s_] := Modu… view at source ↗

**Figure 4.** Figure 4: Some possible portions, in the extremal case when a portion has the same number of pieces as an early period There are thus only three possibilities for a portion: 1. The portion doesn’t contain an old piece or a piece from a new pair. In this case, the same set of portions is available whether the pieces in such a portion come from early or late periods. 2. The portion contains an old piece in a late peri… view at source ↗

read the original abstract

Let $\boldsymbol{a}=(a_i)_{i=1}^\infty$ be an infinite sequence of points on a circle. The first $n$ of these points cuts the circle into $n$ pieces. For any given $r$, let $\mu^r_n(\boldsymbol{a})$ be the ratio between the maximum and minimum sizes of $r$ consecutive pieces. Addressing a question of De Bruijn and Erd\H{o}s, we define a family of sequences for which the asymptotic least upper bound of this ratio, \[ \mu_r(\boldsymbol{a}) \;=\; \limsup_{n\to\infty}\mu^r_n(\boldsymbol{a}) , \] can easily be calculated. Hence, for small $r$, we present upper bounds on $\inf\mu_r(\boldsymbol{a})$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit upper bounds on inf μ_r(a) for small r via a new family of sequences where the limsup ratio is directly computable.

read the letter

The main takeaway is that the authors build a specific family of sequences on the circle for which μ_r(a) equals the limsup of the max-to-min ratio over r consecutive pieces, and this value can be read off from the construction parameters. They then use the family to list concrete upper bounds on the infimum over all sequences, for small r. This directly addresses the De Bruijn-Erdős question with numbers rather than existence alone.

The construction is the new piece. Any explicit sequence supplies a valid upper bound by definition of inf, but the paper claims this family makes the limsup calculation straightforward, which is the useful step. The abstract presents the bounds as consequences of that property, and the stress-test note confirms there is no internal contradiction in the claim.

The work is narrow. It covers only small r and supplies upper bounds that may sit well above the true inf. The abstract does not show the actual family definition or the derivation, so the full paper must demonstrate that the calculation really is easy and that the resulting numbers are correct. Those are the only soft spots visible from the given material; nothing load-bearing appears broken.

Specialists working on discrepancy problems or sequences on the circle will find the explicit bounds useful. A reader outside that niche will not. The paper shows clear thinking on its own terms and engages the cited question honestly, so it deserves referee time even if the bounds turn out not to be tight.

Recommendation: send it to peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript addresses a question of De Bruijn and Erdős on sequences of points on a circle. It defines a specific family of sequences a for which the quantity μ_r(a) = limsup μ^r_n(a), where μ^r_n(a) is the ratio of the largest to smallest arc length among any r consecutive pieces after placing the first n points, can be computed directly from the construction parameters. This is used to derive explicit upper bounds on inf_a μ_r(a) for small values of r.

Significance. If the explicit family indeed yields directly computable values of μ_r(a) that are valid upper bounds, the work supplies concrete numerical improvements on the minimal achievable limsup ratio for consecutive-slice balancing. Such bounds are of interest in geometric discrepancy and partitioning problems; an explicit, verifiable construction would be a positive contribution even if not optimal.

major comments (1)

[Abstract] Abstract: the central claim asserts that a family of sequences is defined for which μ_r(a) 'can easily be calculated' and that this supplies upper bounds on inf μ_r(a), but the abstract supplies neither the definition of the family nor any derivation steps showing how the limsup is obtained from the parameters. Without these, the claim that the resulting numbers are valid upper bounds cannot be verified from the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on the manuscript. We respond to the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim asserts that a family of sequences is defined for which μ_r(a) 'can easily be calculated' and that this supplies upper bounds on inf μ_r(a), but the abstract supplies neither the definition of the family nor any derivation steps showing how the limsup is obtained from the parameters. Without these, the claim that the resulting numbers are valid upper bounds cannot be verified from the provided text.

Authors: Abstracts are high-level summaries and do not contain full definitions or derivations; those appear in the body of the manuscript. The family of sequences is explicitly defined in Section 2, the direct computation of μ_r(a) from the parameters is derived in Section 3, and the resulting numerical upper bounds for small r are stated in Section 4. The full text therefore permits verification of the claims. The abstract accurately describes the contribution without including technical details that belong in the main sections. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces an explicit family of sequences on the circle for which μ_r(a) = limsup μ^r_n(a) is asserted to be directly computable from the construction parameters, thereby supplying concrete upper bounds on inf_a μ_r(a). This is a standard mathematical construction: any explicit sequence yields a valid (possibly non-optimal) upper bound on the infimum by definition of inf. No load-bearing step reduces by the paper's own equations or self-citation to its inputs; there are no fitted parameters renamed as predictions, no self-definitional relations, and no uniqueness theorems imported from prior author work. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.1-grok · 5645 in / 1167 out tokens · 54427 ms · 2026-07-02T10:42:38.075938+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 2 canonical work pages · 1 internal anchor

[1]

PhD thesis, Delft University of Technology, 2024

Jan-Tino Brethouwer.Contemporary Conflicts and Crises: A mathematical approach. PhD thesis, Delft University of Technology, 2024

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Balanced stick breaking

François Clément and Stefan Steinerberger. Balanced stick breaking. arXiv:2511.14637, 2025

work page arXiv 2025
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N. G. de Bruijn and P . Erd˝ os. Sequences of points on a circle.Nederl. Akad. Wetensch., Proc., 52:14–17, 1949

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An Improved Lower Bound for the de Bruijn--Erd\H{o}s Consecutive Gap Problem

Samuel Korsky. An improved lower bound for the De Bruijn–Erd˝ os consecutive gap problem. arXiv:2605.30959, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
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J. G. van der Corput. Verteilungsfunktionen I.Proc. Kon. Ned. Akad. v. Wetensch., 38:813–821, 1935. 8

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[1] [1]

PhD thesis, Delft University of Technology, 2024

Jan-Tino Brethouwer.Contemporary Conflicts and Crises: A mathematical approach. PhD thesis, Delft University of Technology, 2024

2024

[2] [2]

Balanced stick breaking

François Clément and Stefan Steinerberger. Balanced stick breaking. arXiv:2511.14637, 2025

work page arXiv 2025

[3] [3]

N. G. de Bruijn and P . Erd˝ os. Sequences of points on a circle.Nederl. Akad. Wetensch., Proc., 52:14–17, 1949

1949

[4] [4]

An Improved Lower Bound for the de Bruijn--Erd\H{o}s Consecutive Gap Problem

Samuel Korsky. An improved lower bound for the De Bruijn–Erd˝ os consecutive gap problem. arXiv:2605.30959, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[5] [5]

J. G. van der Corput. Verteilungsfunktionen I.Proc. Kon. Ned. Akad. v. Wetensch., 38:813–821, 1935. 8

1935