35.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 35

References:

erdosproblems.com/35
[Er56] Erdős, P., Problems and results in additive number theory. Colloque sur la Théorie des Nombres, Bruxelles, 1955 (1956), 127-137.
[Er36c] Erdős, P., On the arithmetical density of the sum of two sequences, one of which forms a basis for the integers. Acta. Arith. (1936), 197-200.
[Pl70] Plünnecke, H., Eine zahlentheoretische Anwendung der Graphentheorie. J. Reine. Angew. Math. (1970), 171-183.

Imports (1)

FormalConjecturesUtil

theorem Erdos35.erdos_35 sorry line 57 source
depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Membership.mem, OfNat.ofNat, Set.IsAddBasisOfOrder, LE.le, Real, HAdd.hAdd, schnirelmannDensity, Classical.propDecidable, HDiv.hDiv, HMul.hMul, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos35.erdos_35.variants.plunnecke Documented declaration: plunnecke sorry line 68 source
Source documentation
Plünnecke [Pl70] proved that if $B\subseteq\mathbb{N}$ is an additive basis of order $k$ with $0\in B$ then $d_S(A+B)\geq d_s(A)^{1-1/k}$ for every $A\subseteq\mathbb{N}$ with $d_s(A)>0$.

depends on sorry what this means
Statement uses: Set, Nat, Membership.mem, OfNat.ofNat, Set.IsAddBasisOfOrder, LT.lt, Real, schnirelmannDensity, Classical.propDecidable, LE.le, HPow.hPow, HSub.hSub, Inv.inv, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_35.variants.plunnecke (B : Set ℕ) (k : ℕ) (h0 : 0 ∈ B) (hB : B.IsAddBasisOfOrder k)
    (A : Set ℕ) (hA : 0 < schnirelmannDensity A) :
    schnirelmannDensity A ^ (1 - (k : ℝ)⁻¹) ≤ schnirelmannDensity (A + B) := by
…
```