1. FormalConjectures
  2. GreensOpenProblems
  3. «47»
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Green's Open Problem 47

References:

    [Gr24] Ben Green's 100 Open Problems

    [GH14] Green, Ben, and Adam J. Harper. "Inverse questions for the large sieve." Geometric and Functional Analysis 24.4 (2014): 1167-1203.

    [HV09] Helfgott, Harald Andrés, and Akshay Venkatesh. "How small must ill-distributed sets be." Analytic number theory 2 (2009): 224-234.

    [Wa12] Walsh, Miguel N. "The inverse sieve problem in high dimensions." (2012): 2001-2022.

    [Wa14] Walsh, Miguel N. "The algebraicity of ill-distributed sets." Geometric and Functional Analysis 24.3 (2014): 959-967.

open Filter namespace Green47

Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?

The following very particular instance is probably the simplest [Gr24]: Suppose that $A \subset \mathbb{N}$ is a set with the property that $|A \pmod p| \leqslant \frac{1}{2}(p + 1)$ for all sufficiently large $p$. Is it true that either $|A \cap [X]| \ll X^{1/2} / \log^{100} X$, or $A$ is contained in the image of $\mathbb{Z}$ under a quadratic map $\phi : \mathbb{Q} \to \mathbb{Q}$?

@[category research open, AMS 11] theorem declaration uses 'sorry'green_47 : answer(sorry) ↔ ∀ A : Set ℕ, (∀ᶠ p in atTop, Nat.Prime p → Set.ncard (Set.image (fun a : ℕ => (a : ZMod p)) A) ≤ (p + 1) / 2) → ((fun X : ℕ => ((A ∩ Set.Iic X).ncard : ℝ)) ≪ (fun X : ℕ => Real.sqrt (X : ℝ) / (Real.log (X : ℝ)) ^ 100)) ∨ (∃ P : Polynomial ℚ, P.degree = 2 ∧ ∀ a ∈ A, ∃ z : ℤ, (a : ℚ) = P.eval (z : ℚ)) := ⊢ True ↔ ∀ (A : Set ℕ), (∀ᶠ (p : ℕ) in atTop, Nat.Prime p → ((fun a => ↑a) '' A).ncard ≤ (p + 1) / 2) → ((fun X => ↑(A ∩ Set.Iic X).ncard) =O[atTop] fun X => √↑X / Real.log ↑X ^ 100) ∨ ∃ P, P.degree = 2 ∧ ∀ a ∈ A, ∃ z, ↑a = Polynomial.eval (↑z) P All goals completed! 🐙 end Green47