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Let have at least
elements, for some
(we say that
has density at least
), and very large
.
We know, by Meshulam’s theorem (Roth’s theorem in vector spaces), and even more strongly by Ellenberg-Gijswijt, who used the Croot-Lev-Pach method, that must contain a 3-term arithmetic progression, in other words an affine line.
But does it have to contain planes or even affine subspaces of larger dimensions?
First we examine the case of a random set , thus each
is taken in
with probability
independently of each other.
This implies that any given subspace has probability
to be included in
.
Now how many affine subspaces of dimension are there in a space of dimension
? This is
where
is the number of linear subspaces of dimension
, and is easily seen to equal
One gets the obvious bound
Thus the expected number of subspaces of dimension in
is at most
.
This is of the order of magnitude of a constant
when is of the order of
.
So when , there may be no single subspace of dimension
.
However, using
the lower bound
may be used to show similarly that when
the set is likely (in fact, almost sure) to contain subspaces of dimension
.
Now we prove that if a set has positive density and is large enough, then
must contain a subspace of dimension
.
The key is basically to use Varnavides averaging argument along with the Ellenberg-Gijswijt bound.
Let be the minimum
(if it exists) such that any set
of density
must contain a subspace of dimension
. The Ellenberg-Gijswijt bound amounts to
; indeed,
just has to be at least
to ensure a line. Then Varnavides argument is an averaging trick that says that a dense set must contain many lines. So many that it must contain many parallel lines. So many that the set of starting points of these lines must in turn contain a line. In which case we obtain a plane. And so on. This was used by Brown and Buhler to derive the recursive relationship
where and
is the number of linear spaces of dimension
in a space of dimension
. We can bound
by
, which yields
.
And so
Inducting, one obtains
Thus . In other words
, which is exactly the claimed result.
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