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.

theorem Green47.green_47 : sorry ↔ ∀ (A : Set ℕ), (∀ᶠ (p : ℕ) in Filter.atTop, Nat.Prime p → ((fun (a : ℕ) => ↑a) '' A).ncard ≤ (p + 1) / 2) → ((fun (X : ℕ) => ↑(A ∩ Set.Iic X).ncard) =O[Filter.atTop] fun (X : ℕ) => √↑X / Real.log ↑X ^ 100) ∨ ∃ (P : Polynomial ℚ), P.degree = 2 ∧ ∀ a ∈ A, ∃ (z : ℤ), ↑a = Polynomial.eval (↑z) P

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 $\phi : \mathbb{Q} \to \mathbb{Q}$?