Raimi's theorem for manifolds with circle symmetry

Dung The Tran

arxiv: 2512.08676 · v2 · submitted 2025-12-09 · 🧮 math.CO · math.CA· math.NT

Raimi's theorem for manifolds with circle symmetry

Dung The Tran This is my paper

Pith reviewed 2026-05-16 23:58 UTC · model grok-4.3

classification 🧮 math.CO math.CAmath.NT

keywords Raimi's theoremmeasurable partitionscircle symmetryunavoidable setssurface measuresmanifoldsrotational surfacesbundle structures

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The pith

Raimi's unavoidability property lifts from the circle to spheres, cones, and cylinders.

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

This paper shows that three families of surfaces with rotational symmetry—the unit sphere, rotational power surfaces like cones and paraboloids, and circular cylinders—carry measurable partitions with a strong unavoidability feature. For any finite coloring of the surface, there exists a rotation such that the rotated partition intersects a color class in positive measure on both sides. The construction relies on viewing these surfaces as circle bundles over a base space, where the known partition on the circle group can be lifted using the product structure and measure disintegration. A sympathetic reader would see this as extending classical partition theorems from number theory and groups to geometric settings that possess a hidden circle symmetry.

Core claim

The central claim is that for surfaces admitting a measurable trivialization as C × Y with a measure-preserving circle action by rotation, the Raimi partition on the circle lifts to an unavoidable partition on the surface, as verified for the sphere, power surfaces, and cylinders through equivariance and disintegration properties.

What carries the argument

The circle-bundle structure consisting of a measurable product trivialization C × Y together with the measure-preserving rotation action of the circle group.

If this is right

The unavoidability property holds for the natural surface measure on the unit sphere in R^n.
It holds for rotational power surfaces such as cones and paraboloids.
It holds for circular cylindrical surfaces.
A general circle-bundle theorem unifies the proofs for all three families.

Where Pith is reading between the lines

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

This lifting technique may apply to other manifolds with similar circle actions, such as tori or higher-dimensional rotationally symmetric objects.
Connections could be drawn to ergodic theory, where such partitions relate to invariant measures and mixing properties.
Potential extensions to non-measurable versions or other group actions beyond the circle.

Load-bearing premise

The surface must admit a measurable trivialization as a product of the circle and a base space such that the surface measure disintegrates appropriately along the circle fibers.

What would settle it

Constructing a finite measurable coloring of the sphere or a cylinder for which no rotation makes the image intersect both parts of the proposed partition in positive measure would falsify the result.

read the original abstract

Raimi's classical theorem establishes a partition of the natural numbers with a remarkable unavoidability property: for every finite coloring of $\mathbb{N}$, there is a color class whose translate meets both parts of the partition in infinitely many points. Recently, Kang, Koh, and Tran have extended this phenomenon to the circle group, proving that there exists a measurable partition of the circle such that every finite measurable cover admits a rotation whose image meets each part of the partition in positive measure. This paper shows that this phenomenon extends beyond compact abelian groups to a wide class of non-group geometric surfaces that still exhibit \textit{a hidden one-dimensional symmetry}. Specifically, we establish analogs of Raimi's theorem for three families of surfaces (with their natural surface measures): the unit sphere $S^{n-1} \subset \mathbb{R}^n$, rotational power surfaces (such as cones and paraboloids), and circular cylindrical surfaces. The common feature is that each of these surfaces carries a natural measure-preserving action of the circle group by rotation in a fixed plane and admits a measurable trivialization as a product $C \times Y$. This circle-bundle structure allows the measurable Raimi partition on the base circle to be lifted to an unavoidable partition on the manifold. Our approach is unified through a general circle-bundle theorem, which reduces all three geometric cases to verifying suitable equivariance and product disintegration properties of the surface measure.

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

This paper lifts the circle version of Raimi's theorem to spheres, power surfaces, and cylinders by reducing each to a measurable circle-bundle trivialization.

read the letter

The main point is that the authors extend the recent measurable Raimi partition on the circle to three families of surfaces that carry a rotational circle action: the unit sphere, rotational power surfaces such as cones and paraboloids, and circular cylinders. They do this by establishing a general circle-bundle theorem that lets them pull the partition back once the surface measure disintegrates as a product with respect to the circle fibers and the action is equivariant and measure-preserving. Fixed-point sets have measure zero, which lets them get the product structure almost everywhere. This is the actual new step beyond the group case cited in the abstract. What works is the unification. Instead of separate arguments for each surface, the paper isolates the common symmetry and reduces the problem to checking the trivialization and disintegration properties. That keeps the argument short and focused on the geometry that matters. The sphere and cylinder cases look routine, and the power surfaces fit the same pattern once the natural measure is handled correctly. The soft spot is the level of detail on those verifications. The abstract states that equivariance and product disintegration hold for all three families, but the explicit constructions and any measure-theoretic controls are not visible here. If the full paper supplies careful checks, especially for the non-flat power surfaces, this is minor. If those steps are only sketched, the central claim rests on unshown calculations. There is no circularity or self-referential fitting. This is for people working in measurable dynamics who want to see how combinatorial partition properties move from groups to symmetric manifolds. A reader already familiar with the circle case will see the value in the lifting method. It deserves a serious referee because the reduction is logical and the claim is concrete enough to be checked directly.

Referee Report

2 major / 2 minor

Summary. The paper extends Raimi's theorem on unavoidable partitions from the natural numbers and the circle group to three families of surfaces equipped with natural surface measures and a circle-group action by rotation in a fixed plane: the unit sphere S^{n-1} in R^n, rotational power surfaces (cones, paraboloids), and circular cylinders. The central argument introduces a general circle-bundle theorem that reduces each geometric case to the existence of a measurable trivialization as C × Y together with equivariance of the surface measure under the circle action and its disintegration with respect to the circle fibers; the measurable Raimi partition on the base circle is then lifted to an unavoidable partition on the total space.

Significance. If the technical verifications hold, the work supplies a unified, measure-theoretic mechanism for transporting combinatorial unavoidability results across manifolds that possess a hidden one-dimensional rotational symmetry. The reduction of three distinct geometric settings to a single circle-bundle statement is a clear organizational strength and could serve as a template for analogous extensions to other homogeneous spaces or surfaces with compact group actions.

major comments (2)

[General circle-bundle theorem] The general circle-bundle theorem (stated in the abstract and presumably proved in §2) asserts that the lifted partition inherits the unavoidability property once equivariance and product disintegration are verified, yet the manuscript provides no explicit construction of the measurable section, no error-control estimates for the measure-zero fixed-point sets, and no derivation of the positive-measure intersection property after lifting. These steps are load-bearing for the central claim.
[Sphere case] For the unit sphere S^{n-1} (treated in the first geometric case), the disintegration of surface measure with respect to the circle fibers is asserted but not accompanied by an explicit formula or verification that the induced measure on the base is equivalent to Lebesgue measure on the circle; without this, the lifting of the Kang-Koh-Tran partition cannot be confirmed to preserve positive-measure intersections.

minor comments (2)

[Introduction] The definition of 'rotational power surfaces' (cones, paraboloids, etc.) should be stated with an explicit equation or parametrization in §1 to make the circle action and the product structure C × Y unambiguous.
[Notation] Notation for the circle group action and the measurable trivialization map should be introduced once and used consistently; currently the abstract and the geometric sections employ slightly varying symbols for the same objects.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and will revise the paper accordingly to strengthen the explicit details in the proofs.

read point-by-point responses

Referee: [General circle-bundle theorem] The general circle-bundle theorem (stated in the abstract and presumably proved in §2) asserts that the lifted partition inherits the unavoidability property once equivariance and product disintegration are verified, yet the manuscript provides no explicit construction of the measurable section, no error-control estimates for the measure-zero fixed-point sets, and no derivation of the positive-measure intersection property after lifting. These steps are load-bearing for the central claim.

Authors: We appreciate the referee's identification of these load-bearing steps. In the revised manuscript we will expand §2 to include: (i) an explicit construction of the measurable section obtained directly from the given circle-bundle trivialization C × Y; (ii) a short argument showing that the fixed-point sets of the circle action have surface measure zero (by equivariance and Fubini); and (iii) a derivation of the positive-measure intersection property obtained by integrating the indicator functions of the lifted sets over the fibers and invoking the product disintegration together with the Kang-Koh-Tran property on the base. These additions will make the lifting argument fully rigorous without altering the overall strategy. revision: yes
Referee: [Sphere case] For the unit sphere S^{n-1} (treated in the first geometric case), the disintegration of surface measure with respect to the circle fibers is asserted but not accompanied by an explicit formula or verification that the induced measure on the base is equivalent to Lebesgue measure on the circle; without this, the lifting of the Kang-Koh-Tran partition cannot be confirmed to preserve positive-measure intersections.

Authors: We agree that an explicit formula and equivalence verification are required for complete rigor. In the revision we will add the explicit disintegration formula for the surface measure on S^{n-1} (using the standard spherical coordinates with respect to the fixed rotation plane) and prove that the induced measure on the base circle is mutually absolutely continuous with Lebesgue measure, with Radon-Nikodym derivative bounded between two positive constants independent of n. This equivalence guarantees that the positive-measure intersections on the base lift to positive-measure intersections on the sphere, confirming the unavoidability property. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to circle base case; lifting theorem is independent

full rationale

The derivation proceeds by establishing a general circle-bundle theorem that lifts any measurable Raimi partition from the base circle C to the manifold via measurable trivialization as C × Y and disintegration of surface measure with respect to the circle fibers. This reduces the three geometric cases (sphere, rotational power surfaces, cylinders) to verifying equivariance and product disintegration, which are checked directly from the geometry and do not depend on the specific partition chosen on C. The citation to Kang, Koh, and Tran for the circle result is a standard reference to prior independent work and is not load-bearing for the manifold extension. No equations equate a new claim to a fitted parameter or prior result by construction, and no ansatz is smuggled via self-citation. The argument remains self-contained against the external benchmark of the circle theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a measurable product disintegration of the surface measure with respect to the circle action; this is treated as a standard property of the listed surfaces rather than derived inside the paper.

axioms (2)

domain assumption The surface measure on each listed manifold is invariant under the natural circle rotation and admits a measurable trivialization as product measure on C times Y.
Invoked to lift the base-circle partition; appears in the description of the unified circle-bundle theorem.
standard math Standard facts from measurable dynamics and disintegration of measures on circle bundles hold for these surfaces.
Background measure theory used without proof.

pith-pipeline@v0.9.0 · 5541 in / 1401 out tokens · 29467 ms · 2026-05-16T23:58:13.874982+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

general circle-bundle theorem... measurable bijection Φ : C×Y → X∖N satisfying equivariance and product disintegration
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

lifts the measurable Raimi partition on the base circle

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Bergelson and B

V. Bergelson and B. Weiss, Translation Properties of Sets of Positive Upper Density, Proceedings of the American Mathematical Society, 94(3) (1985), 371--376

work page 1985
[2]

Bergelson and A

V. Bergelson and A. Leibman, Polynomial extensions of Van der Waerden's and Szemer\' e di's theorems , Journal of the American Mathematical Society, 9 (1996), 725--753

work page 1996
[3]

Hegyv\'ari, On intersecting properties of partitions of integers, Combinatorics, Probability and Computing, 14(3) (2005), 319--323

N. Hegyv\'ari, On intersecting properties of partitions of integers, Combinatorics, Probability and Computing, 14(3) (2005), 319--323

work page 2005
[4]

Hegy\-v\' a ri, J

N. Hegy\-v\' a ri, J. Pach, and T. Pham, Polynomial extensions of Raimi's theorem, arXiv:2511.06650 [math.CO] https://arxiv.org/abs/2511.06650, (2025)

work page arXiv 2025
[5]

Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc

N. Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), pp. 119--184, Lecture Notes in Math., 751, Springer, Berlin, 1979

work page 1979
[6]

H. Kang, D. Koh, and D. T. Tran, Raimi's theorem for the n -dimensional torus, https://arxiv.org/abs/2512.00935 arXiv:2512.00935 [math.CO] , (2025)

work page arXiv 2025
[7]

Mattila, A survey on the Hausdorff dimension of intersections, Mathematical and Computational Applications, 28(2) (2023), 49

P. Mattila, A survey on the Hausdorff dimension of intersections, Mathematical and Computational Applications, 28(2) (2023), 49

work page 2023
[8]

Raimi, Translation properties of finite partitions of the positive integers, Fundamenta Mathematicae, 61(3) (1967), 253--256

R. Raimi, Translation properties of finite partitions of the positive integers, Fundamenta Mathematicae, 61(3) (1967), 253--256

work page 1967

[1] [1]

Bergelson and B

V. Bergelson and B. Weiss, Translation Properties of Sets of Positive Upper Density, Proceedings of the American Mathematical Society, 94(3) (1985), 371--376

work page 1985

[2] [2]

Bergelson and A

V. Bergelson and A. Leibman, Polynomial extensions of Van der Waerden's and Szemer\' e di's theorems , Journal of the American Mathematical Society, 9 (1996), 725--753

work page 1996

[3] [3]

Hegyv\'ari, On intersecting properties of partitions of integers, Combinatorics, Probability and Computing, 14(3) (2005), 319--323

N. Hegyv\'ari, On intersecting properties of partitions of integers, Combinatorics, Probability and Computing, 14(3) (2005), 319--323

work page 2005

[4] [4]

Hegy\-v\' a ri, J

N. Hegy\-v\' a ri, J. Pach, and T. Pham, Polynomial extensions of Raimi's theorem, arXiv:2511.06650 [math.CO] https://arxiv.org/abs/2511.06650, (2025)

work page arXiv 2025

[5] [5]

Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc

N. Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), pp. 119--184, Lecture Notes in Math., 751, Springer, Berlin, 1979

work page 1979

[6] [6]

H. Kang, D. Koh, and D. T. Tran, Raimi's theorem for the n -dimensional torus, https://arxiv.org/abs/2512.00935 arXiv:2512.00935 [math.CO] , (2025)

work page arXiv 2025

[7] [7]

Mattila, A survey on the Hausdorff dimension of intersections, Mathematical and Computational Applications, 28(2) (2023), 49

P. Mattila, A survey on the Hausdorff dimension of intersections, Mathematical and Computational Applications, 28(2) (2023), 49

work page 2023

[8] [8]

Raimi, Translation properties of finite partitions of the positive integers, Fundamenta Mathematicae, 61(3) (1967), 253--256

R. Raimi, Translation properties of finite partitions of the positive integers, Fundamenta Mathematicae, 61(3) (1967), 253--256

work page 1967