Gauss circle problem: exponent 209/670, below 5/16
Awaiting review
Importance: major (4 of 5)The takeaway
The error term in counting lattice points in a circle gets its first exponent below 5/16, a threshold Huxley had suggested might be the limit of the Bombieri–Iwaniec method; the authors disclose ChatGPT and Codex help with strategy and checking.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 17 October 2026
Your AI and this story
- GPT-6 Astra169 days after its cutoff
- Claude Opus 5.5108 days after its cutoff
- Gemini 4 Argon199 days after its cutoff
- Grok 4.7138 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 108 days before it.
Key facts
- Paper: ‘A New Bound for the Gauss Circle Problem’, Xi Cen, Xitao Gao, Cheng Zhang & Junyong Zhang, arXiv 2610.21730 [math.NT]
- Result: |E(X)| ≲_ε X^(209/670+ε) for all real X ≥ 4, where E(X) = #{(a,b) ∈ ℤ² : a² + b² ≤ X} − πX; 209/670 = 5/16 − 3/5360
- Previous records named in the paper: Gauss O(X^(1/2)); Huxley X^(131/416)(log X)^(18637/8320) (≈ 0.31490); Li–Yang 0.3144831759741…, using Guth–Maldague small-cap decoupling; Bourgain–Watt linked the first-spacing problem to decoupling. Conjectured: X^(1/4+ε)
- Method: a cone distribution estimate from Guth–Maldague’s amplitude-dependent decoupling that keeps fourth- and sixth-power contributions separate in the double large sieve, and an explicit inversion of Taylor-coefficient constraints that cancels the linear error in a pair-counting argument; plus Robert–Sargos and Watt spacing estimates and the Vinogradov mean value theorem
- AI disclosure (quoted): ‘During the preparation of this manuscript, the authors used ChatGPT and Codex to discuss possible proof strategies, assist in drafting and checking parts of the mathematical arguments, organize references, and improve the clarity and readability of the exposition. All mathematical statements and proofs in the final manuscript were written, reviewed, and verified by the authors.’ No model version is named
- No Hacker News thread or press coverage found by Oct 17
What happened
The preprint appeared in arXiv’s number-theory listing on 16 Oct 2026. It refines the Bombieri–Iwaniec/Huxley method using Guth–Maldague decoupling, building on Li and Yang’s recent improvement, and gets the exponent 209/670. The authors present 5/16 as a historical threshold: Huxley had suggested it as a possible limit of the method, and Popov and Li–Yang repeated the remark.
Why it matters
The Gauss circle problem is a textbook open problem, and exponent records move by tiny amounts every decade or so. Crossing 5/16 is symbolically important. The AI role is stated broadly (strategy discussion, drafting and checking parts of the arguments) without saying which ideas came from the models, so the result should be read as human-led with AI assistance until experts check it.
What is disputed or not yet verified
| Verification | Unrefereed preprint; no formal proof |
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Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
- arXiv 2610.21730: A New Bound for the Gauss Circle Problemarxiv.org, paper
Changes
- Filed (arXiv /list/math/new page for 16 Oct 2026; PDF read with pdftotext)