Ben Green's Open Problem 7

Does Ulam's sequence have positive density? Can one explain the curious Fourier properties of Ulam's sequence?

References:

• Ben Green's Open Problem 7
• erdosproblems.com/342

theorem Green7.green_7.variants.positive_density : True ↔ ∀ (a : ℕ → ℕ), Erdos342.IsUlamSequence a → (Set.range a).upperDensity > 0

Does Ulam's sequence have positive density?