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Radius of comparison equals half the mean dimension
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Abstract
We prove the integer-action case of the Phillips–Toms conjecture: for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of its crossed product equals one half of its mean dimension, including equality at infinity. For these systems, zero mean dimension and the small boundary property are equivalent to Jiang–Su stability and to finite nuclear dimension of the crossed product; in this case its nuclear dimension is at most one.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- George A. Elliott and Zhuang Niu, Mean dimension and radius of comparison (arXiv:2610.11143v1, 2026-10-08). This follow-up, whose authors disclose ChatGPT use, adapts the comparison and cover-reconstruction arguments using the companion's torus-compression theorem to obtain equality for diagonal AH algebras and free minimal amenable actions with both the uniform Rokhlin property and Cuntz comparison of open sets. [family-level]
Related prior literature mentioned by others
- George A. Elliott and Zhuang Niu, On the small boundary property and $\mathcal Z$-absorption (arXiv:2504.03611v3, revised 2026-08-13). Cited in the October follow-up, this revision characterizes the small boundary property through property (C) and leaves its implication from Jiang–Su absorption open, providing context for the release paper's regularity corollary.
Possibly relevant literature (found by the agent, not checked)
- Zhuang Niu, Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C*-algebras (arXiv:1906.09172, v1 June 2019, v2 August 2020). Proves rc(C(X)⋊Γ) ≤ ½ mdim under a uniform Rokhlin property and an open-set comparison condition, both stated to hold for Γ = ℤ; it is a companion to the paper's cited source for the known upper bound (Niu, Canad. J. Math. 2024).
- Taylor Hines, The radius of comparison and mean dimension (Purdue University PhD dissertation, ProQuest AAI3735175). Proposes that the mean dimension of a minimal system equals twice the radius of comparison of its crossed product and gives supporting examples; this is the identity the paper's Theorem 1.1 asserts for minimal homeomorphisms.
- M. Ali Asadi-Vasfi and N. Christopher Phillips, The Dynamical Radius of Comparison for C*-Dynamical Systems (arXiv:2609.30211, 24 September 2026). Its introduction states the Phillips–Toms conjecture rc = ½ mdim for free minimal actions of countable amenable groups, the broader formulation the paper says lies outside its scope (it treats ℤ-actions only).
Digests and expositions
- AI Proved What?, Exactly one half? The Phillips–Toms conjecture (Episode 302, October 2026). This AI-assisted video and transcript explain both papers' claims, the role of cobordism compression in reducing cover order, and the Elliott–Niu follow-up. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 302
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026,
author = {{OpenAI}},
title = {{Radius of comparison equals half the mean dimension}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026/paper.pdf}{OAI:Radius-of-comparison-equals-half-the-mean-dimension-September-25-2026}},
year = {2026}
}Family 302: Radius of comparison equals half the mean dimension
For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of $C(X)\rtimes_h\mathbb Z$ equals $\tfrac12\mathrm{mdim}(X,h)$, including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.
Other papers in this family:
- Filtered products and boundary-preserving compression in complex cobordism · September 25, 2026