この記事の日本語版はまだありません。英語版を表示しています。

Littlewood polynomials with large normalized Mahler measure

An internal OpenAI model produced a preprint proving that real ultraflat Littlewood polynomials exist, which settles an Epoch AI open problem; the proof has not yet been independently checked.

問題
Is there an infinite family of polynomials with coefficients ±1\pm 1 whose Mahler measure, divided by n+1\sqrt{n+1}, stays above 0.970.97?
貢献
OpenAI internal model · Part of OpenAI's October 2026 release of results
検証
Lean formalization supplied by OpenAIMarked solved by Epoch AI
経緯
Previous record 0.954 → OpenAI preprint dated 2026-10-05, released 2026-10-06
出典
Epoch AI problem pageOpenAI preprint
未検証Lean formalization by OpenAI · no independent check

The problem

A Littlewood polynomial has every coefficient equal to +1+1 or −1-1. Epoch AI's FrontierMath Open Problems asks for an infinite family of them whose Mahler measure, normalized by n+1\sqrt{n+1}, stays above 0.970.97. The best value previously known was 0.9540.954.

What AI did

As part of the batch of results OpenAI released on 6 October 2026 (result family 076), an internal OpenAI model produced a preprint proving that real ultraflat Littlewood polynomials exist. That implies normalized Mahler measure arbitrarily close to 11, which is far more than the problem asked for. Epoch AI marked the problem as solved by AI on the strength of the preprint.

How it was checked

OpenAI links a Lean formalization from its results repository. We found no independent check of the proof, and it has not been peer reviewed.

Limitations

Like the rest of OpenAI's October release, the model that produced the result is not public, so the work cannot be reproduced. Lean certifies only the formal statement it was given; whether that statement matches the paper's claim needs its own check. Problem description paraphrased from Epoch AI (CC BY 4.0).

∎