61.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

1 declarations, 1 contain sorry, 0 declared as axioms

Ben Green's Open Problem 61

Reference: [Ben Green's Open Problem 61](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 61)

This problem was originally considered by Erdős and Newman.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem Green61.green_61 Suppose that $A + A$ contains the first $n$ squares sorry line 42 source
Source documentation
Suppose that $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1 - o(1)}$?
It is known that necessarily $|A| \geq n^{2/3 - o(1)}$, whilst in the other direction there do exist such $A$ with $|A| \ll_C n / \log^C n$ for any $C$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, GE.ge, Finset, LE.le, Finset.image, HPow.hPow, Finset.Icc, HAdd.hAdd, HSub.hSub, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
56 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_61 :
    answer(sorry) ↔
      ∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧
        ∀ n : ℕ, n ≥ 1 → ∀ (A : Finset ℕ),
          (Finset.Icc 1 n).image (· ^ 2) ⊆ A + A →
            (n : ℝ) ^ (1 - f n) ≤ A.card := by
…
```