498.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 498

References:

erdosproblems.com/498
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Er45] Erdős, P., On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. (1945), 898-902.
[Kl65] Kleitman, Daniel J., On a lemma of Littlewood and Offord on the distribution of certain sums. Math. Z. (1965), 251-259.
[Kl70] Kleitman, Daniel J., On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors. Advances in Math. (1970), 155-157.

Imports (1)

FormalConjecturesUtil

theorem Erdos498.erdos_498 Let $z_1,\ldots,z_n\in\mathbb{C}$ with $1\leq \lvert z_i\rvert$ for $1\leq i\leq n$ sorry line 53 source
Source documentation
Let $z_1,\ldots,z_n\in\mathbb{C}$ with $1\leq \lvert z_i\rvert$ for $1\leq i\leq n$. Let $D$ be an arbitrary disc of radius $1$. Is it true that the number of sums of the shape $$\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\}$$ which lie in $D$ is at most $\binom{n}{\lfloor n/2\rfloor}$?
A strong form of the Littlewood-Offord problem. Erdős [Er45] proved this is true if $z_i\in\mathbb{R}$, and for general $z_i\in\mathbb{C}$ proved a weaker upper bound of $$\ll \frac{2^n}{\sqrt{n}}.$$ This was solved in the affirmative by Kleitman [Kl65], who also later generalised this to arbitrary Hilbert spaces [Kl70].
See also [395].

depends on sorry what this means
Statement uses: Iff, True, Nat, Fin, Complex, LE.le, Real, OfNat.ofNat, Norm.norm, Set.ncard, Int, Set.ofPred, And, Or, Eq, Neg.neg, Membership.mem, Set, Metric.ball, Finset.sum and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_498 : answer(True) ↔
    ∀ (n : ℕ) (z : Fin n → ℂ), (∀ i, 1 ≤ ‖z i‖) → ∀ c : ℂ,
      {ε : Fin n → ℤ | (∀ i, ε i = -1 ∨ ε i = 1) ∧
        (∑ i, (ε i : ℂ) * z i) ∈ Metric.ball c 1}.ncard ≤ n.choose (n / 2) := by
…
```
Mentions: c