116.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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 116

References:

erdosproblems.com/116
[EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67.
[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.
[KLR25] M. Krishnapur, E. Lundberg, and K. Ramachandran, On the area of polynomial lemniscates. arXiv:2503.18270 (2025).
[Wa88] Wagner, Gerold, On the area of lemniscate domains. J. Analyse Math. (1988), 159-167.
[Po28] G. Pólya, Beitrag zue Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete. S-B. Akad. Wiss. (1928), 228-232 and 280-282.

Imports (1)

FormalConjecturesUtil

theorem Erdos116.erdos_116 sorry line 69 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Fin, Complex, LE.le, Norm.norm, ENNReal, ENNReal.ofReal, HDiv.hDiv, HPow.hPow, MeasureTheory.Measure, Set, MeasureTheory.MeasureSpace.volume, Set.ofPred and 3 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 Erdos116.erdos_116.variants.log The lower bound sorry line 80 source
Source documentation
The lower bound $\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \gg (\log n)^{-1}$ was proved by Krishnapur, Lundberg, and Ramachandran [KLR25]; in particular the answer to the $(\log n)^{-O(1)}$ form of the question is also yes.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Fin, Complex, Norm.norm, ENNReal, ENNReal.ofReal, HDiv.hDiv, Real.log, MeasureTheory.Measure, Set, MeasureTheory.MeasureSpace.volume, Set.ofPred, Finset.prod, Finset.univ and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
58 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_116.variants.log : ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 2 ≤ n →
    ∀ z : Fin n → ℂ, (∀ i, ‖z i‖ ≤ 1) →
      ENNReal.ofReal (c / Real.log n) ≤ volume {w : ℂ | ‖∏ i, (w - z i)‖ < 1} := by
…
```
Mentions: c
theorem Erdos116.erdos_116.variants.upper Documented declaration: upper sorry line 91 source
Source documentation
Krishnapur, Lundberg, and Ramachandran [KLR25] proved that there are polynomials with $\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \ll (\log\log n)^{-1}$, improving a result of Wagner [Wa88].

depends on sorry what this means
Statement uses: Exists, Real, Nat, LE.le, OfNat.ofNat, Fin, Complex, And, Norm.norm, ENNReal, MeasureTheory.Measure, Set, MeasureTheory.MeasureSpace.volume, Set.ofPred, LT.lt, Finset.prod, Finset.univ, HSub.hSub, ENNReal.ofReal, HDiv.hDiv and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_116.variants.upper : ∃ C : ℝ, ∀ n : ℕ, 3 ≤ n →
    ∃ z : Fin n → ℂ, (∀ i, ‖z i‖ ≤ 1) ∧
      volume {w : ℂ | ‖∏ i, (w - z i)‖ < 1} ≤ ENNReal.ofReal (C / Real.log (Real.log n)) := by
…
```
theorem Erdos116.erdos_116.variants.polya Pólya [Po28]: $\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \leq \pi$ always holds sorry line 98 source
Source documentation
Pólya [Po28]: $\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \leq \pi$ always holds.

depends on sorry what this means
Statement uses: Nat, Fin, Complex, LE.le, ENNReal, MeasureTheory.Measure, Set, MeasureTheory.MeasureSpace.volume, Set.ofPred, LT.lt, Real, Norm.norm, Finset.prod, Finset.univ, HSub.hSub, OfNat.ofNat, ENNReal.ofReal, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
39 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_116.variants.polya (n : ℕ) (z : Fin n → ℂ) :
    volume {w : ℂ | ‖∏ i, (w - z i)‖ < 1} ≤ ENNReal.ofReal Real.pi := by
…
```