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Ben Green's Open Problem 76: green_76.variants.weak_net

+170 XP to settle

Added 1 Nov 2026 · not yet checked against the literature · importance 0.50

What the prompt says

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Statement

Do weak $\varepsilon$ -nets of size $O(1/\varepsilon)$ exist for axis-parallel rectangles in the plane? [Gr24] believes that green_76 is an instance of this question, which he states for axis-parallel boxes. With $k = \varepsilon n$ : is there a constant $C$ such that, for every $k \ge 1$ , every set $A$ of $n$ points in the plane has a set $A'$ of at most $Cn/k$ points that meets every axis-parallel rectangle containing at least $k$ points of $A$ ?

A positive answer implies green_76.variants.exists_constant, and it implies green_76 if the constant $C$ is at most $500$ . [AES10, Theorem 2.2] gives such sets of size $O(\varepsilon^{-1} \log \log \varepsilon^{-1})$ , even with $A' \subseteq A$ . When $A' \subseteq A$ is required, this size is optimal [PT13, Theorem 4].

Lean statement

theorem green_76.variants.weak_net :
    ∃ C : ℝ, ∀ k : ℕ, 0 < k → ∀ A : Finset (ℝ × ℝ),
      ∃ A', #A' ≤ C * #A / k ∧ HitsRichRectangles k A A' := by
  sorry

Sources(1)

formal-conjectures: FormalConjectures/GreensOpenProblems/76.lean (opens in a new tab)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/GreensOpenProblems/76.lean.

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