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import FormalConjecturesUtilBen Green's Open Problem 61
This problem was originally considered by Erdős and Newman.
open scoped Pointwise Topologyopen Filter
namespace Green61
Suppose that $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1 - o(1)}$?
It is known that necessarily $|A| \geq n^{2/3 - o(1)}$, whilst in the other direction there do exist such $A$ with $|A| \ll_C n / \log^C n$ for any $C$.
@[category research open, AMS 11]
theorem green_61 :
answer(sorry) ↔
∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧
∀ n : ℕ, n ≥ 1 → ∀ (A : Finset ℕ),
(Finset.Icc 1 n).image (· ^ 2) ⊆ A + A →
(n : ℝ) ^ (1 - f n) ≤ A.card := ⊢ True ↔
∃ f,
Tendsto f atTop (𝓝 0) ∧
∀ n ≥ 1, ∀ (A : Finset ℕ), Finset.image (fun x => x ^ 2) (Finset.Icc 1 n) ⊆ A + A → ↑n ^ (1 - f n) ≤ ↑A.card
All goals completed! 🐙
end Green61