VibeMathedMath problems solved by AI

Brezis's Open Problem 5.6 on Universal Fourier Summation

Does a universal summation process recover the degree of a circle map from its Fourier moduli, that is, does ∑nσn,εn∣f^(n)∣2→deg⁡f\sum_n \sigma_{n,\varepsilon} n |\hat f(n)|^2 \to \deg f hold for Holder maps below the threshold? No. For every 0<α<1/30 < \alpha < 1/3 there is an f∈C0,α(S1;S1)f \in C^{0,\alpha}(S^1;S^1) for which the sum fails to converge to deg⁡f\deg f, answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all p>3p > 3. The endpoint C0,1/3C^{0,1/3} is left unresolved.

Result
Disproved(see note)
Status
Partial result
AI contribution
AI co-developed
Method
Construction
Field
Harmonic analysis
Posed by
Haim Brezis
Year posed
—
Years open
—
Solved
2026-07-26
Model
ChatGPT
Vendor
OpenAI
Collaborators
Michal Cieszynski
Verification
Unreviewed
Publication
Preprint
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

negative below the 1/3 threshold; the endpoint case is still open

What the AI did

The disclosure states that ChatGPT generated preliminary drafts of the proofs in the manuscript. The author then verified each argument in detail, revised the proofs where necessary, checked the cited sources, determined the final formulation of all results, and takes sole responsibility for the content.

Verification

Single-author arXiv preprint; the construction combines degree-zero quotients of Blaschke factors with a Baire category argument. Not yet peer-reviewed.

Source

arXiv:2607.23598 - Nonexistence of universal Fourier summation formulas for the degree below the Holder threshold

Discussion