33.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, 1 contain sorry, 0 declared as axioms

Ben Green's Open Problem 33

References:

[Gr24] Ben Green's Open Problem 33
[CaHa20] Caprace, Pierre-Emmanuel, and Pierre de la Harpe. "Groups with irreducibly unfaithful subsets for unitary representations." Confluentes Mathematici 12.1 (2020): 31-68.
[CrLe07] Croot, Ernie, and Vsevolod F. Lev. "Open problems in additive combinatorics." Additive combinatorics 43.207-233 (2007): 1.

Imports (1)

FormalConjecturesUtil

theorem Green33.green_33 Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$? sorry line 43 source
Source documentation
Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$? [Gr24]

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Filter.Frequently, PNat, Exists, Finset, ZMod, And, Eq, HAdd.hAdd, Finset.univ, abs, HSub.hSub, HDiv.hDiv, Finset.card, Real.sqrt, Nat and 1 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 green_33 :
    answer(sorry) ↔
      ∀ ε : ℝ, 0 < ε →
        ∃ᶠ q : ℕ+ in atTop,
          ∃ A : Finset (ZMod q),
            A + A = Finset.univ ∧
            |((A.card : ℝ) / Real.sqrt q - Real.sqrt 2)| < ε := by
…
```
theorem Green33.green_33.sanity_sq_bound Trivial lower bound: if $A + A = \mathbb{Z}/q\mathbb{Z}$, then $|A|^2 \geq q$, since the sumset $A + A$ has at most $|A|^2$ elements line 55 source
Source documentation
Trivial lower bound: if $A + A = \mathbb{Z}/q\mathbb{Z}$, then $|A|^2 \geq q$, since the sumset $A + A$ has at most $|A|^2$ elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PNat, Finset, ZMod, Eq, HAdd.hAdd, Finset.univ, LE.le, Nat, HPow.hPow, Finset.card, OfNat.ofNat
Proof uses: Trans.trans, HMul.hMul, Fintype.card, Eq.symm, ZMod.card, Finset.card_univ, Eq.mpr, id, congrArg, Eq.refl, Finset.card_add_le, sq
41 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem green_33.sanity_sq_bound (q : ℕ+) (A : Finset (ZMod q))
    (hA : A + A = Finset.univ) : (q : ℕ) ≤ A.card ^ 2 := by
…
```