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import FormalConjecturesUtilErdős Problem 245
namespace Erdos245
open Filter Set Erdos245
open scoped Pointwise Topology
Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap {1, ..., N}| = o(N)$. Is it true that $$ \limsup_{N\to\infty}\frac{|(A + A)\cap {1, ..., N}|}{|A \cap {1, ..., N}|} \geq 3? $$
The answer is yes, proved by Freiman [Fr73].
[Fr73] Fre\u{\i}man, G. A.,
@[category research solved, AMS 5 11]
theorem erdos_245 :
answer(True) ↔ ∀ (A : Set ℕ), A.Infinite →
atTop.Tendsto (fun N ↦ (A ∩ Icc 1 ⌊N⌋₊ |>.ncard : ℝ) / N) (𝓝 0) →
3 ≤ atTop.limsup
fun N : ℝ ↦ ((A + A) ∩ Icc 1 ⌊N⌋₊ |>.ncard : EReal)
/ (A ∩ Icc 1 ⌊N⌋₊).ncard := ⊢ True ↔
∀ (A : Set ℕ),
A.Infinite →
Tendsto (fun N => ↑(A ∩ Icc 1 ⌊N⌋₊).ncard / N) atTop (𝓝 0) →
3 ≤ limsup (fun N => ↑((A + A) ∩ Icc 1 ⌊N⌋₊).ncard / ↑(A ∩ Icc 1 ⌊N⌋₊).ncard) atTop
All goals completed! 🐙
Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap {1, ..., N}| = o(N)$. Then $$ \limsup_{N\to\infty}\frac{|(A + A)\cap {1, ..., N}|}{|A \cap {1, ..., N}|} \geq 2. $$
@[category research solved, AMS 5 11]
theorem erdos_245.variants.two (A : Set ℕ) (h_inf : A.Infinite)
(hf : atTop.Tendsto (fun N ↦ (A ∩ Icc 1 ⌊N⌋₊ |>.ncard : ℝ) / N) (𝓝 0)) :
2 ≤ atTop.limsup
fun N : ℝ ↦ ((A + A) ∩ Icc 1 ⌊N⌋₊ |>.ncard : EReal)
/ (A ∩ Icc 1 ⌊N⌋₊).ncard := A:Set ℕh_inf:A.Infinitehf:Tendsto (fun N => ↑(A ∩ Icc 1 ⌊N⌋₊).ncard / N) atTop (𝓝 0)⊢ 2 ≤ limsup (fun N => ↑((A + A) ∩ Icc 1 ⌊N⌋₊).ncard / ↑(A ∩ Icc 1 ⌊N⌋₊).ncard) atTop
All goals completed! 🐙
end Erdos245