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.

theorem Green15.green_15 : sorry ↔ ∃ (K : NNReal) (f : ℕ → ℤ), LipschitzWith K f ∧ {x : ℤ × ℤ | ∃ (n : ℕ), (↑n, f n) = x}.IsAPOfLengthFree 3

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?

theorem Green15.green_15_ap4 : ∃ (K : NNReal) (f : ℕ → ℤ), LipschitzWith K f ∧ {x : ℤ × ℤ | ∃ (n : ℕ), (↑n, f n) = x}.IsAPOfLengthFree 4

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