230.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 230

References:

erdosproblems.com/230
[Er57] Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300.
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.
[Er80h] Erdős, P., Problems and results on polynomials and interpolation. (1980), 383--391.
[Ko80] Körner, T. W., On a polynomial of Byrnes. Bull. London Math. Soc. (1980), 219--224.
[Ka80] Kahane, Jean-Pierre, Sur les polynômes à coefficients unimodulaires. Bull. London Math. Soc. (1980), 321-342.
[BoBo09] Bombieri, Enrico and Bourgain, Jean, On Kahane's ultraflat polynomials. J. Eur. Math. Soc. (JEMS) (2009), 627-703.

Imports (1)

FormalConjecturesUtil

def Erdos230.circleMax The maximum of $\lvert P(z)\rvert$ over the unit circle, for $P(z)=\sum_{1\leq k\leq n}a_kz^k$ line 45 source
Source documentation
The maximum of $\lvert P(z)\rvert$ over the unit circle, for $P(z)=\sum_{1\leq k\leq n}a_kz^k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Complex, Real
Proof uses: iSup, Metric.sphere, OfNat.ofNat, Norm.norm, Finset.sum, Finset.univ, HMul.hMul, HPow.hPow, Membership.mem, Set, HAdd.hAdd
27 instances, coercions and auxiliary constants left out.
```
noncomputable def circleMax {n : ℕ} (a : Fin n → ℂ) : ℝ :=
…
```
theorem Erdos230.erdos_230 sorry line 76 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Fin, Complex, Eq, Norm.norm, HMul.hMul, HAdd.hAdd, Real.sqrt, Erdos230.circleMax
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos230.erdos_230.variants.ultraflat Kahane [Ka80] constructed 'ultraflat' polynomials sorry line 86 source
Source documentation
Kahane [Ka80] constructed 'ultraflat' polynomials $P_n(z)=\sum_{k \leq n} a_kz^k$ with $\lvert a_k\rvert=1$ such that $P_n(z)=(1+o(1))\sqrt{n}$ uniformly for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$.

depends on sorry what this means
Statement uses: Exists, Nat, Fin, Complex, And, Eq, Real, Norm.norm, OfNat.ofNat, Filter.Tendsto, iSup, Metric.sphere, abs, HSub.hSub, HDiv.hDiv, Finset.sum, Finset.univ, HMul.hMul, HPow.hPow, Membership.mem and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
82 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_230.variants.ultraflat : ∃ a : (n : ℕ) → Fin n → ℂ, (∀ n k, ‖a n k‖ = 1) ∧
    Tendsto (fun n : ℕ => ⨆ z : Metric.sphere (0 : ℂ) 1,
      |‖∑ k : Fin n, a n k * (z : ℂ) ^ ((k : ℕ) + 1)‖ / √n - 1|) atTop (𝓝 0) := by
…
```