512.lean`

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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

Erdős Problem 512

References:

erdosproblems.com/512
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Ko81] Konyagin, S. V., On the Littlewood problem. Izv. Akad. Nauk SSSR Ser. Mat. (1981), 243-265, 463.
[MPS81] McGehee, O. Carruth and Pigno, Louis and Smith, Brent, Hardy's inequality and the $L^1$ norm of exponential sums. Ann. of Math. (2) (1981), 613-618.

Imports (1)

FormalConjecturesUtil

theorem Erdos512.erdos_512 Documented declaration: erdos_512 sorry line 48 source
Source documentation
Is it true that, if $A\subset \mathbb{Z}$ is a finite set of size $N$, then $$\int_0^1 \left\lvert \sum_{n\in A}e(n\theta)\right\rvert \mathrm{d}\theta \gg \log N,$$ where $e(x)=e^{2\pi ix }$?
Littlewood's conjecture, proved independently by Konyagin [Ko81] and McGehee, Pigno, and Smith [MPS81].

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Finset, Int, Eq, Finset.card, LE.le, HMul.hMul, Real.log, intervalIntegral, Norm.norm, Complex, Finset.sum, additiveChar and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
41 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_512 : answer(True) ↔
    ∃ c > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℤ), A.card = N →
      c * Real.log N ≤ ∫ θ in (0 : ℝ)..1, ‖∑ n ∈ A, e (n * θ)‖ := by
…
```
Mentions: c