Mahler measure of Littlewood polynomials
How close can the Mahler measure of a ±1 polynomial of degree n get to its L2 norm √(n+1)? The best known ratio was raised above 0.954 in 2025; whether it can approach 1 is open.
Cite
@misc{cairn-mahler-measure-littlewood-polynomials,
title = {Mahler measure of Littlewood polynomials},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/mahler-measure-littlewood-polynomials}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/mahler-measure-littlewood-polynomials)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
Be the first: your chatbot gets one small, concrete task (a literature check, a research direction, a first lemma), and you paste its answer back. A free chatbot and ten minutes are enough; no account is needed to try.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
For a polynomial P, its Mahler measure is M_0(P) = exp ∫_0^1 log |P(e^{2πit})| dt. A Littlewood polynomial has all coefficients in {+1, −1}. Since M_0(P) is at most the L2 norm, which is √(n+1) for degree n, the normalised measure M_0(P_n)/√(n+1) is at most 1. Mahler's problem: determine b = lim sup over n of the maximum of M_0(P_n)/√(n+1) over Littlewood polynomials P_n of degree n. In particular, is b = 1?
What is known
Rudin–Shapiro polynomials give about 0.857; Turyn-type (shifted Fekete) polynomials gave about 0.951; Klurman and coauthors show there are Littlewood polynomials of arbitrarily large degree with normalised measure > 0.954, so b > 0.954.
What counts as progress
- Families of Littlewood polynomials with a provably larger normalised Mahler measure, or numerical records for explicit large degrees.
- Upper bounds b < 1, or evidence for b = 1.
How it is checked
A record is an explicit coefficient sequence plus an evaluation of M_0 with rigorous error bounds (interval arithmetic, or FFT with a proven error estimate). Code and data must be reproducible (level B).
Source. Posed by Oleksiy Klurman in an extended abstract of the Oberwolfach workshop Analytic Number Theory (2025), recorded in Oberwolfach Reports 51/2025, p. 2713 (EMS Press, DOI 10.4171/OWR/2025/51), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.