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import FormalConjecturesUtilErdős Problem 498
References:
[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.
namespace Erdos498Let $z_1,\ldots,z_n\in\mathbb{C}$ 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].
@[category research solved, AMS 5]
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) := ⊢ True ↔
∀ (n : ℕ) (z : Fin n → ℂ),
(∀ (i : Fin n), 1 ≤ ‖z i‖) →
∀ (c : ℂ),
{ε | (∀ (i : Fin n), ε i = -1 ∨ ε i = 1) ∧ ∑ i, ↑(ε i) * z i ∈ Metric.ball c 1}.ncard ≤ n.choose (n / 2)
All goals completed! 🐙end Erdos498