Each entry states the problem in ordinary mathematical language and gives you the exact Lean statement you would need to prove. Nothing is paraphrased, so what you read is what gets checked.
Combinatorics
Green50.green_50
Let A⊂F2n be a set of density α>0. Does 10A
contain a coset
of some subspace of dimension at least
n−O(log(1/α))
?
More precisely: does there exist an absolute constant
C>0
such that for all
n≥1
and all
nonempty
A⊆F2n
with density
α>0
, the sumset
10A
contains a coset
of some subspace of dimension at least
n−Clog2(1/α)
?
The sumset
10A
is defined as
{a1+a2+⋯+a10:ai∈A}
, using the pointwise
scalar multiplication notation
10 • A
where
•
denotes the iterated addition of a set.
Note: We model
F2n
as
Fin n → ZMod 2
, which is an
n
-dimensional vector space
over
F2
.
References
Ben Green's Open Problem 50
No one has attempted this yet.
Formal statement
Lean type
True ↔
∃ C > 0,
∀ (n : ℕ) (A : Finset (𝔽₂ n)),
A.Nonempty →
have α := ↑A.dens;
∃ W v, v +ᵥ ↑W ⊆ ↑(10 • A) ∧ ↑n - C * Real.logb 2 (1 / α) ≤ ↑(Module.finrank (ZMod 2) ↥W)
What you must prove
import FormalConjectures.GreensOpenProblems.«50»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Green50.green_50") := by
sorry
end Bounty
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.