OpenFrontiers
Join
← Back to the index
OP-9372OpenAnalysis

Erdős #1150 · analysis / polynomials

Does there exist a constant c>0c>0 such that, for all large nn and all polynomials PP of degree nn with coefficients ±1\pm 1,

max⁡∣z∣=1∣P(z)∣>(1+c)n?\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?
Source reports OPENAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Does there exist a constant c>0c>0 such that, for all large nn and all polynomials PP of degree nn with coefficients ±1\pm 1,

max⁡∣z∣=1∣P(z)∣>(1+c)n?\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?

Lean 4

No Lean submitted

Formal statement

The formalization is available at the source below; its code is not attached here yet.

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.

Proofs & verification

No Lean has been submitted for this problem yet. A Lean file can be attached to a submission for review, though with no target here it cannot be machine-checked.

Share a formalization →Verification guide ↗

Background

In other words, does there exist an 'ultraflat' polynomial with coefficients ±1\pm 1. The answer is yes if the coefficients can take any values on the unit circle (see [230]).

The lower bound

max⁡∣z∣=1∣P(z)∣≥n\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].

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1150 link
  2. [2]Upstream Lean 4 statement for Erdős #1150 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1150 link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.