FormalConjectures.GreensOpenProblems.«15»

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import FormalConjecturesUtil

Green's Open Problem 15

References:

[Gr24] Green, Ben. "100 open problems." (2024).

[BJP14] T. Brown, V. Jungić and A. Poelstra, "On double 3-term arithmetic progressions", Integers 14 (2014), Paper No. A43.

[CCS14] J. Cassaigne, J. D. Currie, L. Schaeffer and J. Shallit, "Avoidance of additive cubes and related results", Adv. in Appl. Math. 56 (2014), 25–66.


open Set

open scoped NNReal


namespace Green15

Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = {(n, f(n)) : n \in \mathbb{N}} \subseteq \mathbb{Z}^2$ is free of 3-term progressions?

@[category research open, AMS 5 11]
theorem declaration uses 'sorry'green_15 :
    answer(sorry) ↔ ∃ K : ℝ≥0, ∃ f : ℕ → ℤ, LipschitzWith K f ∧
      IsAPOfLengthFree {((n, f n) : ℤ × ℤ) | (n : ℕ)} 3 := by⊢ True ↔ ∃ K f, LipschitzWith K f ∧ {x | ∃ n, (↑n, f n) = x}.IsAPOfLengthFree 3
  sorryAll goals completed! 🐙

The answer is YES for 4-term progressions [BJP14].

@[category research solved, AMS 5 11]
theorem declaration uses 'sorry'green_15_ap4 :
    ∃ K : ℝ≥0, ∃ f : ℕ → ℤ, LipschitzWith K f ∧
      IsAPOfLengthFree  {((n, f n) : ℤ × ℤ) | (n : ℕ)} 4 := by⊢ ∃ K f, LipschitzWith K f ∧ {x | ∃ n, (↑n, f n) = x}.IsAPOfLengthFree 4
  sorryAll goals completed! 🐙


-- TODO(jeangud) Add finitary version

end Green15