395.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, 3 contain sorry, 0 declared as axioms

Erdős Problem 395

References:

erdosproblems.com/395
[Er45] Erdős, P., On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. (1945), 898--902.
[CaCa11] Carnielli, Walter and Carolino, Pietro K., Adjusting a conjecture of Erdős. Contrib. Discrete Math. (2011), 154--159.
[HJNS24] X. He, T. Juškevičius, B. Narayanan, and S. Spiro, The Reverse Littlewood-Offord problem of Erdős. arXiv:2408.11034 (2024).

Imports (1)

FormalConjecturesUtil

def Erdos395.signedSumCount The number of sign patterns $\epsilon \in \{-1,1\}^n$ with $\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq r$ line 39 source
Source documentation
The number of sign patterns $\epsilon \in \{-1,1\}^n$ with $\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq r$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Complex, Real
Proof uses: Set.ncard, Int, Set.ofPred, And, Or, Eq, Neg.neg, OfNat.ofNat, LE.le, Norm.norm, Finset.sum, Finset.univ, HMul.hMul
10 instances, coercions and auxiliary constants left out.
```
noncomputable def signedSumCount {n : ℕ} (z : Fin n → ℂ) (r : ℝ) : ℕ :=
…
```
theorem Erdos395.erdos_395 sorry line 59 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Fin, Complex, Eq, Norm.norm, LE.le, HDiv.hDiv, Erdos395.signedSumCount, Real.sqrt, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos395.erdos_395.variants.one Documented declaration: one sorry line 69 source
Source documentation
Erdős originally asked erdős_395 with $\sqrt{2}$ replaced by $1$, but Carnielli and Carolino [CaCa11] observed that this is false, choosing $z_1=1$ and $z_k=i$ for $2\leq k\leq n$, where $n$ is even, since then the sum is at least $\sqrt{2}$ always.

L4DEL explanation
Erdős originally asked erdős_395 with $\sqrt{2}$ replaced by $1$, but Carnielli and Carolino [CaCa11] observed that this is false, choosing $z_1=1$ and $z_k=i$ for $2\leq k\leq n$, where $n$ is even, since then the sum is at least $\sqrt{2}$ always.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Fin, Complex, Eq, Norm.norm, LE.le, HDiv.hDiv, Erdos395.signedSumCount, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_395.variants.one : answer(False) ↔ ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 0 < n →
    ∀ z : Fin n → ℂ, (∀ i, ‖z i‖ = 1) → c / n ≤ (signedSumCount z 1 : ℝ) / 2 ^ n := by
…
```
Mentions: c, Erdos395.signedSumCount
theorem Erdos395.erdos_395.variants.sharp The bound of $1/n$ in erdős_395 is the best possible, as shown by taking $z_k=1$ for $1\leq k\leq n/2$ and $z_k=i$ otherwise sorry line 78 source
Source documentation
The bound of $1/n$ in erdős_395 is the best possible, as shown by taking $z_k=1$ for $1\leq k\leq n/2$ and $z_k=i$ otherwise.

L4DEL explanation
The bound of $1/n$ in erdős_395 is the best possible, as shown by taking $z_k=1$ for $1\leq k\leq n/2$ and $z_k=i$ otherwise.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Exists, Real, Nat, LT.lt, OfNat.ofNat, Fin, Complex, And, Eq, Norm.norm, LE.le, HDiv.hDiv, Erdos395.signedSumCount, Real.sqrt, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_395.variants.sharp : ∃ C : ℝ, ∀ n : ℕ, 0 < n → ∃ z : Fin n → ℂ,
    (∀ i, ‖z i‖ = 1) ∧ (signedSumCount z √2 : ℝ) / 2 ^ n ≤ C / n := by
…
```
Mentions: Erdos395.signedSumCount