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Combinatorics
Green's open problem 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
.
$2,690 bounty·nobody has started
week of 27 Sept 2025 · nothing publishedweek of 4 Oct 2025 · nothing publishedweek of 11 Oct 2025 · nothing publishedweek of 18 Oct 2025 · nothing publishedweek of 25 Oct 2025 · nothing publishedweek of 1 Nov 2025 · nothing published
Solve it
Nobody has attempted this. A complete proof or refutation takes the whole bounty.
27 Sept 2025nothing published against this one yet19 Sept 2026
Ben Green's Open Problem 50
Something wrong with this formalization?
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.
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