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mo271 merged 1 commit into from
Apr 4, 2026
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feat(GreenOpenProblems): 7, positive density part #3661

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24 changes: 21 additions & 3 deletions FormalConjectures/GreensOpenProblems/7.lean
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Expand Up @@ -14,14 +14,32 @@ See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjectures.ErdosProblems.«341»
import FormalConjectures.ErdosProblems.«342»

/-!
# 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](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.1)
- [erdosproblems.com/341](https://www.erdosproblems.com/341)
- [erdosproblems.com/342](https://www.erdosproblems.com/342)
-/

open Nat Set Filter
open scoped Topology

This file points to the canonical formalization in `FormalConjectures.ErdosProblems.«341»`.
namespace Green7

/--
Does Ulam's sequence have positive density?
-/
@[category research open, AMS 11 42]
theorem green_7.variants.positive_density :
answer(sorry) ↔
∀ a : ℕ → ℕ, Erdos342.IsUlamSequence a →
Set.upperDensity (Set.range a) > 0 := by
sorry

end Green7
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