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Littlewood Polynomials with Large Normalized Mahler Measure

Littlewood Polynomials with Large Normalized Mahler Measure. Moderately interesting problem from Epoch AI’s FrontierMath: Open Problems; answers are checked by a verifier program.

MSC 30C10 · Polynomials and rational functions of one complex variableMSC 11C08 · Polynomials in number theory

Epoch field label “Analysis” mapped to MSC2020 30C10 for Littlewood Polynomials with Large Normalized Mahler Measure. https://epoch.ai/frontiermath/open-problems/large-mahler-measure Not an author-supplied code.

Licensed source statementAs of Epoch AI data archive updated 2026-10-07; status as labeled by EpochSource & evidence ↓
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Problem statement

For a polynomial P∈C[x],P\in\mathbb{C}[x], the LλL_{\lambda} norm of PP for λ>0\lambda > 0 is

∥P∥λ=(∫01∣P(e(t))∣λdt)1/λ,\|P\|_{\lambda} = \left(\int_0^1 |P(e(t))|^{\lambda} d t\right)^{1/\lambda},

and the Mahler measure of PP is

M0(P)=exp⁡(∫01log⁡∣P(e(t))∣dt).M_0(P)=\exp\left(\int_{0}^1\log |P(e(t))|dt\right).

This problem, known as the Borwein-Mahler problem, asks for an infinite family of polynomials Pn(x)=∑k=0nakxkP_n(x)=\sum_{k=0}^na_kx^k, n→∞n \to \infty, with ak=±1a_k=\pm 1 for all 0≤k≤n0\le k\le n, such that the normalized Mahler measure is large for each polynomial. This benchmark specifically asks for an infinite family of polynomials such that for each PnP_n, it holds that

b(Pn)=M0(Pn)n+1>0.97.b(P_n)=\frac{M_0(P_n)}{\sqrt{n+1}} > 0.97.

Polynomials of this type, i.e. polynomials where each coefficient is ±1\pm 1, are known as Littlewood polynomials. A solution to this problem would likely involve ideas across analysis, combinatorics, and number theory, which were all involved in the current normalized Mahler measure record of 0.954. If a polynomial family were found with normalized Mahler measure arbitrarily close to one, this would shed light on the related important question about ultraflat Littlewood polynomials.

This verifier samples a finite set of nn from the claimed infinite family of polynomials, and the Mahler measure computation involves the possibility of numerical error (since it uses FFT), so the verifier is not exact.

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Background

FrontierMath: Open Problems lists this problem with notability “Moderately interesting”, field “Analysis” and type “Construction - Finite; Improved Bound”. Each problem has a bespoke verifier program, so a proposed answer can be checked automatically.

Verifiable task — Epoch prompt, verbatim

For a polynomial P(x) = sum_{k=0}^d a_k x^k, write e(t) = exp(2*pi*i*t) and recall the Mahler measure

    M_0(P) = exp( integral_0^1 log|P(e(t))| dt ).

A Littlewood polynomial is one whose coefficients all satisfy a_k = +-1. For such a polynomial of degree d, Parseval gives ||P||_2 = sqrt(d+1), and Jensen's inequality yields M_0(P) <= sqrt(d+1). Define the normalized Mahler measure

    b(P) = M_0(P) / sqrt(d+1),

so that 0 < b(P) <= 1. The goal is to construct Littlewood polynomials whose normalized Mahler measure is as large as possible at arbitrarily large degree.

Concretely: find an algorithm that takes a positive integer n as input and produces a Littlewood polynomial of degree strictly greater than n whose normalized Mahler measure satisfies

    b(P) > 0.97.

Solution format:
* Write a Python script defining a function `solution(n: int) -> str`.
* The returned string must be over the alphabet {'+', '-'} and have length d+1 > n+1, where the k-th character (0-indexed) is the sign of the coefficient a_k, listing a_0, a_1, ..., a_d (lowest-order coefficient first). '+' denotes the coefficient +1 and '-' denotes -1.
* Do not include any code at the file level. You may include a `main` block for testing, but it will not be executed by the verifier.
* You may assume n <= 500000; no larger value of n is tested.
* For each tested n the call to solution(n) must complete within 5 minutes on a typical laptop, and the returned degree must not exceed 2000000. Since n is at most 500000, this cap always leaves room for a valid answer.
* An example showing the required format is provided in the file "/solution_example.py".

Progress & source assessment

Epoch marks this problem solved (AI) on 2026-10-06. This is Epoch’s classification; the solution has not been independently checked for this record.

Epoch solution update

First solved by: internal OpenAI model

On October 6, 2026, OpenAI released hundreds of new results in mathematics. In one preprint, they prove the existence of ultraflat real Littlewood polynomials. This implies that there are Littlewood polynomials with normalized Mahler measure arbitrarily close to one. Therefore, we are marking this problem as solved by AI.

References
  1. [1]Epoch AI, ‘FrontierMath: Open Problems’. Published online at epoch.ai. Retrieved from https://epoch.ai/frontiermath/open-problems (data updated 7 October 2026). — Littlewood Polynomials with Large Normalized Mahler Measure linkCC BY 4.0 (Epoch AI data and problem pages); attributed reproduction of problem descriptions and verbatim task prompts.

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