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import FormalConjecturesUtilBen Green's Open Problem 60
open scoped Pointwisenamespace Green60
Is there an absolute constant $c > 0$ such that, whenever $A ⊆ \mathbb{N}$ is a set of squares with $|A| ≥ 2$, the sumset $A + A$ satisfies $|A + A| ≥ |A|^{1 + c}$?
@[category research open, AMS 11]
theorem green_60 :
answer(sorry) ↔ ∃ c > (0 : ℝ),
∀ (A : Finset ℕ),
(∀ a ∈ A, IsSquare a) →
2 ≤ A.card →
((A + A).card : ℝ) ≥ (A.card : ℝ) ^ (1 + c) := ⊢ True ↔ ∃ c > 0, ∀ (A : Finset ℕ), (∀ a ∈ A, IsSquare a) → 2 ≤ A.card → ↑(A + A).card ≥ ↑A.card ^ (1 + c)
All goals completed! 🐙
end Green60