Raimi's theorem for manifolds with circle symmetry

Technical Summary: Raimi's Theorem for Manifolds with Circle Symmetry

Problem Statement
The paper addresses the extension of Raimi's classical theorem from the natural numbers and the circle group to a broader class of non-group geometric surfaces. Raimi's original theorem establishes the existence of a partition of the natural numbers $\mathbb{N}$ that is "unavoidable" under finite colorings and shifts. Specifically, for any finite partition of $\mathbb{N}$, there exists a color class and a shift such that the shifted class intersects both parts of Raimi's partition infinitely.

Recent work by Kang, Koh, and the author extended this phenomenon to the circle group $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, proving the existence of a measurable partition of the circle such that every finite measurable cover admits a rotation intersecting each partition element in positive measure. The central problem of this paper is to determine whether this "unavoidability" phenomenon extends beyond compact abelian groups to geometric surfaces that lack a group structure but possess a hidden one-dimensional symmetry (circle action). The paper specifically targets three families of surfaces: the unit sphere $S^{n-1}$, rotational power surfaces (including cones and paraboloids), and circular cylindrical surfaces.

Methodology
The paper employs a unified structural approach based on circle-bundle theory and measure disintegration. The core strategy involves lifting the known Raimi partition from the base circle $\mathbb{T}$ to the higher-dimensional manifold $X$ by exploiting a measurable product structure.

1. Circle-Bundle Framework: The authors establish a general theorem (Theorem 2.1) for a probability space $(X, \mu)$ equipped with a measure-preserving circle action $\{R_\theta\}_{\theta \in \mathbb{T}}$. The theorem requires the existence of:

   • A null set $N \subset X$ where the action degenerates.
   • A measurable bijection $\Phi: \mathbb{T} \times Y \to X \setminus N$ that intertwines the circle action with translation on the first factor (equivariance).
   • A product disintegration property: The surface measure $\mu$ must decompose such that the conditional measures on the fibers (circle orbits) coincide with the normalized Haar measure (Lebesgue measure) on $\mathbb{T}$.

2. Rokhlin Disintegration: The analytic engine of the proof is Rokhlin's disintegration theorem. The authors argue that because the surface measure is invariant under the circle action, the conditional measures on almost every fiber must be the unique translation-invariant probability measure on the circle (Haar measure). This reduces the geometric problem to a combinatorial one on the base circle.

3. Lifting Mechanism:

   • Given a finite measurable cover of the manifold $X$, the authors pull it back to the product space $\mathbb{T} \times Y$.
   • Using a pigeonhole principle on the base circle $Y$, they construct a measurable partition of the base circle.
   • They apply the known Raimi theorem on the circle $\mathbb{T}$ to find a rotation that intersects every part of the base partition in positive measure.
   • By integrating over the fibers and utilizing the equivariance of the map $\Phi$, they demonstrate that this rotation forces the original cover element to intersect every lifted partition component of $X$ in positive measure.

Key Contributions and Results
The paper provides three specific applications of the general circle-bundle theorem, establishing the existence of "unavoidable" measurable partitions for the following surfaces equipped with their natural surface measures:

1. The Unit Sphere ($S^{n-1}$): For $n \ge 3$, the paper constructs a partition of the sphere such that any finite measurable cover admits a rotation in the $(x_1, x_2)$-plane that intersects each partition element in positive measure. The singular set $N$ corresponds to the poles where $x_1 = x_2 = 0$.
2. Rotational Power Surfaces: This class includes surfaces defined by $x_n = |x'|^k$ (where $x' = (x_1, \dots, x_{n-1})$) for $k > 0$. Special cases include cones ($k=1$) and paraboloids ($k=2$). The result holds for these surfaces within a bounded radius $R$, with the singular axis removed.
3. Circular Cylindrical Surfaces: Defined as $x_1^2 + x_2^2 = R^2$ with the remaining coordinates in a bounded Borel set $\Omega$. Since these surfaces have no singular axis, the null set $N$ is empty.

In all cases, the constructed partition $\{E_i\}_{i=1}^r$ satisfies the property that for any finite measurable cover $\{F_m\}_{m=1}^t$, there exists an index $m_0$ and a rotation $R_{\theta_0}$ such that $\mu(R_{\theta_0}(F_{m_0}) \cap E_i) > 0$ for all $i$.

Significance and Claims
The paper claims that the extension of Raimi's theorem to these geometric surfaces is driven by a "hidden" one-dimensional symmetry rather than a full group structure. The authors emphasize that while the problem is formulated geometrically, the essential mechanism is combinatorial. The analytic contribution is limited to the structural decomposition of the measure (via Rokhlin's theorem) to ensure the conditional measures on fibers are Haar measures.

The paper explicitly answers a question raised in previous work [5] regarding the extension from the circle to the unit sphere. It posits that the "unavoidability" phenomenon is robust across any manifold that admits a measure-preserving circle action and a measurable trivialization as a circle bundle. The authors conclude by noting an open question: whether such partitions exist for compact hyperbolic surfaces, which lack circle actions and thus fall outside the current circle-bundle framework. The paper does not claim to solve this open question or propose new applications beyond the theoretical extension of Ramsey-type phenomena to these specific geometric settings.