310.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 310

References:

erdosproblems.com/310
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[LiSa24] Liu, Y. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113 (2024).
[Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).

Imports (1)

FormalConjecturesUtil

theorem Erdos310.erdos_310 sorry line 50 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, LE.le, Finset, Finset.Icc, HMul.hMul, Finset.card, And, Eq, Rat, Finset.sum, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos310.erdos_310.variants.liu_sawhney Documented declaration: liu_sawhney sorry line 61 source
Source documentation
Liu and Sawhney [LiSa24] proved that if $(\log N)^{-1/7+o(1)}\leq \alpha \leq 1/2$ then there is some $S\subseteq A$ such that $\frac{a}{b}=\sum_{n\in S}\frac{1}{n}$ with $a\leq b \leq \exp(O(1/\alpha))$.

depends on sorry what this means
Statement uses: Exists, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HPow.hPow, Real.log, HAdd.hAdd, Neg.neg, HDiv.hDiv, Finset, Finset.Icc, HMul.hMul, Finset.card, And, Real.exp, Eq, Rat and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
66 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_310.variants.liu_sawhney : ∃ C : ℝ, ∀ ε : ℝ, 0 < ε → ∀ᶠ N : ℕ in Filter.atTop,
    ∀ α : ℝ, Real.log N ^ (-(1 / 7 : ℝ) + ε) ≤ α → α ≤ 1 / 2 →
      ∀ A ⊆ Finset.Icc 1 N, α * N ≤ A.card →
        ∃ S ⊆ A, ∃ a b : ℕ, 0 < a ∧ a ≤ b ∧ (b : ℝ) ≤ Real.exp (C / α) ∧
          ∑ n ∈ S, (1 / n : ℚ) = a / b := by
…
```