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import FormalConjecturesUtilGreen's Open Problem 9
References:
[Gr24] Green, Ben. "100 open problems." (2024).
[BlSi20] Bloom, Thomas F., and Olof Sisask. "Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions." arXiv preprint arXiv:2007.03528 (2020).
open Filter
namespace Green9
The quantity $r_k(N)$, defined as the size of the largest subset of ${1, \dots, N}$ without non-trivial $k$-term arithmetic progressions.
noncomputable def r (k N : ℕ) : ℕ := (Finset.Icc 1 N).maxAPFreeCard k
Problem 9 (i): is $r_3(N) \ll N(\log N)^{-10}$?
Solved in [BlSi20].
@[category research solved, AMS 5]
theorem green_9_i :
(fun (N : ℕ) ↦ (r 3 N : ℝ)) ≪ fun (N : ℕ) ↦ (N : ℝ) * (Real.log N) ^ (-10 : ℝ) := ⊢ (fun N => ↑(r 3 N)) =O[atTop] fun N => ↑N * Real.log ↑N ^ (-10)
All goals completed! 🐙
Problem 9 (ii): is $r_5(N) \ll N(\log N)^{-c}$?
@[category research open, AMS 5]
theorem green_9_ii : answer(sorry) ↔
∃ c > (0 : ℝ), (fun (N : ℕ) ↦ (r 5 N : ℝ))
≪ fun (N : ℕ) ↦ (N : ℝ) * (Real.log N) ^ (-c) := ⊢ True ↔ ∃ c > 0, (fun N => ↑(r 5 N)) =O[atTop] fun N => ↑N * Real.log ↑N ^ (-c)
All goals completed! 🐙
Problem 9 (iii): is $r_4(\mathbf{F}_5^n) \ll N^{1-c}$, where $N=5^n$?
@[category research open, AMS 5]
theorem green_9_iii : answer(sorry) ↔
∃ c > (0 : ℝ), (fun (n : ℕ) ↦ ((Finset.univ : Finset (𝔽₅ n)).maxAPFreeCard 4 : ℝ))
≪ fun (n : ℕ) ↦ ((5 : ℝ) ^ n) ^ (1 - c) := ⊢ True ↔ ∃ c > 0, (fun n => ↑(Finset.maxAPFreeCard 4 Finset.univ)) =O[atTop] fun n => (5 ^ n) ^ (1 - c)
All goals completed! 🐙
-- TODO(jeangud): Add some additional bounds from the literature.
end Green9