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import FormalConjecturesUtilBen Green's Open Problem 2
References:
[Gr24] Green, Ben. "100 open problems." (2024).
[Er65] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965.
[Sa21] Sanders, Tom. "The Erdős–Moser Sum-free Set Problem." Canadian Journal of Mathematics 73.1 (2021): 63-107.
[Ru05] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.
[Ch71] S. L. G. Choi. On a combinatorial problem in number theory. Proc. London Math. Soc. (3), 23:629–642, 1971. doi:10.1112/plms/s3-23.4.629.
[BSS00] A. Baltz, T. Schoen, and A. Srivastav. Probabilistic construction of small strongly sum-free sets via large Sidon sets. Colloq. Math., 86(2):171–176, 2000. doi:10.4064/cm-86-2-171-176.
open Filteropen scoped Topology
namespace Green2
We define the construction from [Sa21, p1] as $M(A) := \max {|S| : S \subseteq A \text{ and } (S \hat{+} S) \cap A = \varnothing }$.
def maxRestrictedSumAvoidingSubsetSize (A : Finset ℤ) : ℕ :=
(A.powerset.filter fun S => Disjoint S.restrictedSumset A).sup Finset.card
Let $A \subset \mathbf{Z}$ be a set of $n$ integers. Is there a set $S \subset A$ of size $(\log n)^{100}$ such that the restricted sumset$S \hat{+} S$ is disjoint from $A$?
@[category research open, AMS 11]
theorem green_2 : answer(sorry) ↔
∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →
(maxRestrictedSumAvoidingSubsetSize A : ℝ) ≥ (Real.log n) ^ 100 := ⊢ True ↔ ∀ᶠ (n : ℕ) in atTop, ∀ (A : Finset ℤ), A.card = n → ↑(maxRestrictedSumAvoidingSubsetSize A) ≥ Real.log ↑n ^ 100
All goals completed! 🐙
From [Sa21] it is known that there is always such an S with $|S| \gt (\log |A|)^{1+c}$.
@[category research solved, AMS 11]
theorem green_2_lower_bound_sanders :
∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℤ, A.card = n →
maxRestrictedSumAvoidingSubsetSize A ≥ Real.log n ^ (1 + c) := ⊢ ∃ c > 0,
∀ᶠ (n : ℕ) in atTop, ∀ (A : Finset ℤ), A.card = n → ↑(maxRestrictedSumAvoidingSubsetSize A) ≥ Real.log ↑n ^ (1 + c)
All goals completed! 🐙
From [Er65] it is known that $M(A) \le \frac{1}{3}|A| + O(1)$.
@[category research solved, AMS 11]
theorem green_2_upper_bound_erdos :
∃ C : ℝ, ∀ᶠ n in atTop, ∃ A : Finset ℤ, A.card = n ∧
(maxRestrictedSumAvoidingSubsetSize A : ℝ) ≤ A.card / 3 + C := ⊢ ∃ C, ∀ᶠ (n : ℕ) in atTop, ∃ A, A.card = n ∧ ↑(maxRestrictedSumAvoidingSubsetSize A) ≤ ↑A.card / 3 + C
All goals completed! 🐙
From [Ch71] it is known that $M(A) \le |A|^{2/5 + o(1)}$.
@[category research solved, AMS 11]
theorem green_2_upper_bound_choi :
∃ (o : ℕ → ℝ) (_ : Tendsto o atTop (𝓝 0)), ∀ᶠ n in atTop, ∃ A : Finset ℤ, A.card = n ∧
(maxRestrictedSumAvoidingSubsetSize A : ℝ) ≤ A.card ^ (2 / 5 + o A.card) := ⊢ ∃ o,
∃ (_ : Tendsto o atTop (𝓝 0)),
∀ᶠ (n : ℕ) in atTop, ∃ A, A.card = n ∧ ↑(maxRestrictedSumAvoidingSubsetSize A) ≤ ↑A.card ^ (2 / 5 + o A.card)
All goals completed! 🐙
From [Ru05] the best-known upper bound is $|S| \lt e^{C \sqrt{\log |A|}}$.
@[category research solved, AMS 11]
theorem green_2_upper_bound_ruzsa :
∃ C > (0 : ℝ), ∀ᶠ n in atTop, ∃ A : Finset ℤ, A.card = n ∧
(maxRestrictedSumAvoidingSubsetSize A : ℝ) < Real.exp (C * Real.sqrt (Real.log A.card)) := ⊢ ∃ C > 0,
∀ᶠ (n : ℕ) in atTop, ∃ A, A.card = n ∧ ↑(maxRestrictedSumAvoidingSubsetSize A) < Real.exp (C * √(Real.log ↑A.card))
All goals completed! 🐙
end Green2