355.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 355

References:

erdosproblems.com/355
[DoKo25] W. van Doorn and V. Kovač, Lacunary sequences whose reciprocal sums represent all rationals in an interval. arXiv:2509.24971 (2025).

Imports (1)

FormalConjecturesUtil

theorem Erdos355.erdos_355 Is there a lacunary sequence sorry line 48 source
Source documentation
Is there a lacunary sequence $A\subseteq \mathbb{N}$ (so that $A=\{a_1 < \cdots\}$ and there exists some $\lambda > 1$ such that $a_{n+1}/a_n\geq \lambda$ for all $n\geq 1$) such that $$\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}$$ contain all rationals in some open interval?
Bleicher and Erdős conjectured the answer is no.
In fact the answer is yes, with any lacunarity constant $\lambda\in (1,2)$ (though not $\lambda=2$), as proved by van Doorn and Kova\v{c} [DoKo25].
This was formalized in Lean by van Doorn using Aristotle.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, IsLacunary, Real, LT.lt, Rat, Membership.mem, Set, Set.Ioo, Set.ofPred, Finset, LE.le, Set.range, Eq, Finset.sum, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_355 :
    answer(True) ↔ ∃ A : ℕ → ℕ, IsLacunary A ∧ ∃ u v : ℝ, u < v ∧ ∀ q : ℚ, ↑q ∈ Set.Ioo u v →
      q ∈ {∑ a ∈ A', (1 / a : ℚ) | (A' : Finset ℕ) (_ : ↑A' ⊆ Set.range A)} := by
…
```
Mentions: IsLacunary