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Level A · Machine-checkable Hard Analysis P-erdos-1038

Erdős Problem #1038

What is the infimum of |x ∈ ℝ : |f x| < 1| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-erdos-1038,
  title        = {Erdős Problem #1038},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1038}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1038». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1038.parts.i (n : ℕ) : answer(sorry) =
    ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧
    (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree},
    volume {x | |f.1.eval x| < 1}

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1038. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/1038
  • [Tao25] Tao, Terence. Sublevel Sets of Logarithmic Potentials. Terry Tao’s Blog, Dec. 2025 (https://terrytao.wordpress.com/wp-content/uploads/2025/12/erdos-1038-1.pdf)

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.