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 $\sum_n \sigma_{n,\varepsilon} n |\hat f(n)|^2 \to \deg f$ hold for Holder maps below the threshold? No. For every $0 < \alpha < 1/3$ there is an $f \in C^{0,\alpha}(S^1;S^1)$ for which the sum fails to converge to $\deg f$ , answering Open Problem 5.6 from Brezis's list of favourite open problems negatively for all $p > 3$ . The endpoint $C^{0,1/3}$ 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: 2023
Years open: 3y
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 Not yet peer-reviewed.

Brezis's Open Problem 5.6 on Universal Fourier Summation

What was actually shown

What the AI did

Verification

Sources

Discussion