OPEN
This is open, and cannot be resolved with a finite computation.
Does there exist a constant $c>0$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\pm 1$,\[\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?\]
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
In other words, does there exist an 'ultraflat' polynomial with coefficients $\pm 1$. The answer is yes if the coefficients can take any values on the unit circle (see
[230]).
The lower bound\[\max_{\lvert z\rvert=1}\lvert P(z)\rvert\geq \sqrt{n}\]is trivial from Parseval's theorem.
A weaker 'flatness' question is the subject of
[228].
View the LaTeX source
This page was last edited 23 January 2026. View history
External data from
the database - you can help update this
Formalised statement?
Yes
|
Likes this problem
|
old-bielefelder,
el_Houcein-A
|
|
Interested in collaborating
|
None
|
|
Currently working on this problem
|
el_Houcein-A
|
|
This problem looks difficult
|
None
|
|
This problem looks tractable
|
None
|
|
The results on this problem could be formalisable
|
None
|
|
I am working on formalising the results on this problem
|
None
|
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #1150, https://www.erdosproblems.com/1150, accessed 2026-04-11
