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Level A · Machine-checkable Hard Number theory P-erdos-143

Erdős Problem #143

Does this imply that liminf |A ∩ [1,x]|/x = 0?

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.

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Cite 
@misc{cairn-erdos-143,
  title        = {Erdős Problem #143},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-143}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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Current state

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

The problem

The question

erdos_143.parts.i. Does this imply that

erdos_143.parts.ii. Or

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«143» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_143.parts.i : answer(sorry) ↔ ∀ (A : Set ℝ), WellSeparatedSet A →
    liminf (fun x => (A ∩ (Set.Icc 1 x)).ncard / x) atTop = 0
theorem erdos_143.parts.ii (A : Set ℝ) (h : WellSeparatedSet A) :
    Summable fun (x : A) ↦ 1 / (x * Real.log x)

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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/143. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/143

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.