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Level A · Machine-checkable Hard Number theory P-erdos-332

Erdős Problem #332

Let A⊆ ℕ and D(A) be the set of those numbers which occur infinitely often as a_1 - a_2 with a_1, a_2∈ A. What conditions on A are sufficient to ensure D(A) has bounded gaps? This is formalised here using the answer(sorry) mechanism.

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-332,
  title        = {Erdős Problem #332},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-332}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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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

Let and be the set of those numbers which occur infinitely often as with . What conditions on are sufficient to ensure has bounded gaps?

This is formalised here using the answer(sorry) mechanism. In order to solve this problem one has to provide what the sufficient conditions are, and proof that they imply the desired condition. If the condition is a solution to the problem is up to human judgement.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«332». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_332 (A : Set ℕ) : (answer(sorry) : Set ℕ → Prop) A → HasBoundedGaps (D_A A)

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/332. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/332

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.