Radius of comparison versus mean dimension of crossed products
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For a minimal homeomorphism h of an infinite compact metrizable space X, is the radius of comparison of C(X)⋊ₕZ exactly half the mean dimension of (X,h), including infinite values? A known upper bound leaves the general lower bound open.
Why it matters and what counts as progress
Why it matters: An exact equality would connect an operator-algebra comparison invariant to the topological mean dimension of a dynamical system.
What counts as progress: Prove the lower bound for arbitrary minimal systems or establish equality under weaker cohomological restrictions.
How the problem developed
- 2015 · Conjecture [1]
Taylor Hines Proposed that the mean dimension of a minimal system equals twice the radius of comparison of its crossed product, with examples supporting the relation.
- 2023 · Partial result [2]
Zhuang Niu Established the upper bound rc(C(X)⋊ℤᵈ) ≤ ½ mdim(X,ℤᵈ) for minimal free ℤᵈ-actions; this did not give the matching lower bound in the target ℤ case.
- 25 Sept 2026 · Reported claim [3]
OpenAI The preprint claims equality, including infinite values, for minimal homeomorphisms of infinite compact metrizable spaces, the integer-action case of the Phillips–Toms conjecture.
AI contribution
Radius of comparison equals half the mean dimension
For every minimal homeomorphism h of an infinite compact metrizable space X, the manuscript claims rc(C(X)⋊ₕZ)=mdim(X,h)/2, including infinite values. The new part is the lower bound for every finite open cover.
Sources
- Taylor Hines, The Radius of Comparison and Mean Dimension
- Zhuang Niu, Radius of Comparison and Mean Topological Dimension: ℤᵈ-Actions
- Radius of comparison equals half the mean dimension
- Radius of comparison equals half the mean dimension
- OpenAI mathematics release
- Elliott and Niu, later extension of the comparison method
Website timeline
Publication dates and editorial changes.
OpenAI reports radius of comparison equals half mean dimension