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Sharpness of Denjoy's Theorem

Denjoy's 1932 theorem says a C1+bvC^{1+\mathrm{bv}} circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity ω\omega weaker than Lipschitz, there is a C1+ωC^{1+\omega} circle diffeomorphism with irrational rotation number and a wandering interval. The case ω(t)=tlog⁡(1/t)\omega(t) = t\log(1/t) settles an open problem going back to Herman's 1979 work, which had constructions only for ω(t)=tlog⁡(1/t)1+ε\omega(t) = t\log(1/t)^{1+\varepsilon}.

Result
Proved
Status
Resolved
AI contribution
AI co-developed
Method
Construction
Field
Dynamical systems
Posed by
Michael Herman
Year posed
1979
Years open
47y
Solved
2026-08-03
Model
GPT-5.6 Sol Ultra; Claude Fable 5
Vendor
—
Collaborators
Rohil Prasad
Verification
Unreviewed
Publication
Preprint
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What the AI did

The AI use section says the author prompted GPT-5.6 Sol Ultra to construct a Denjoy example for the modulus tlog⁡(1/t)t\log(1/t), which is the corollary settling Herman's case, and used Claude Fable 5 to search for errors.

Verification

arXiv preprint, not yet peer-reviewed.

Source

arXiv:2608.02380 - On the sharpness of Denjoy's theorem

Discussion