WEBVTT
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YUFEI ZHAO: OK, we are
still on our journey
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to proving Freiman theorem.
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Right?
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So we've been
looking at some tools
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for analyzing sets
of small doubling.
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And last time, we showed
the following result,
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that if a has small doubling,
then there exists a prime.
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It's not too much
bigger than a, such
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that a big subset of a, of
at least 1/8 proportion,
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a big subset of a is Freiman
8-isomorphic to a subset
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of this small cyclic group.
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So last time, we developed this
tool called of modeling lemma,
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so Ruzsa's modeling
lemma that allows
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us to pass from a set of small
doubling, which could have
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elements very spread
out in the integers,
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to something that is much more
compact, a much tighter set.
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That's a subset, a
positive proportionate
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subset of a small cyclic group.
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And remember, the last time we
defined this notion of Freiman
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8-isomorphic, Freiman
isomorphism, in this case,
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it just means that it preserves
partially additive structure.
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It preserves additive
structure when
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you look at most 8-wise sums.
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All right.
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Well, this is where
we left off last time.
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If you start with
small doubling,
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then I can model a big
portion of this set
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by a large fraction of
a small cyclic group.
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All right.
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So now, we're in
the setting where
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we are looking at some space,
a cyclic group, for instance.
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And now, we have
positive proportion,
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a constant proportion
subset of that group.
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And we would like to extract
some additional additive
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structure from this large set.
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And that should
remind you of things
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we've discussed when we
talked about Roth's theorem.
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Right?
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So Roth's theorem
also had this form.
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If you start with Z mod n or
in the finite field setting,
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and you have a constant
proportion of the space,
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then you must find a three-term
arithmetic progression.
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In fact, you must
find many three-APs.
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So we're going to do something
very similar, at least
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in spirit, here.
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We're starting from this large
proportion of some space.
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We're going to extract a very
large additive structure, just
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from the size alone.
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So let me begin by motivating
it with a question.
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And we're going to start,
as we've done in the past,
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with a finite field model, where
things are much easier to state
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and to analyze.
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The question is, suppose
you have a set a, which
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is a subset of f2 to the m.
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And a is an alpha proportion
of the space where you think
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of alpha as some constant.
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Question is, OK,
suppose this is true.
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And must it be the case
that a plus a the subset--
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all right, so a itself,
just because it's
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a large proportion of the space.
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So just because it's
the 1% of the space,
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doesn't mean that it contains
any large structures.
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It doesn't contain necessarily
any large sub-spaces,
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because it could be a more
random subset of f to the m.
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But there's a general principle
in additive combinatorics
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or even analysis where if
you start with a set that
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is quite large, and it might be
a bit rough, a is a bit rough,
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it's all over the place.
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If you add a to itself,
it smooths out the set.
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So a plus a is much
smoother than a.
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And the question is, must A
plus A contain a large subspace?
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And here, by "large,"
I mean the following.
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Because we're looking
at constant proportions
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of the entire
space, by "large," I
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would also want a constant
proportion of the entire space.
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So does there
exist some subspace
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of bounded codimension?
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So if alpha is a
constant, I want
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a bounded codimensional subspace
that lives inside A plus A.
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It turns out the answer is no.
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So there exists sets that, even
though there are very large
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and you add it to
itself, it still
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doesn't have large subspaces.
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So let me give you an example.
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And this construction
is called a nivo set.
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So let's take A sub n to be the
set of points in F2 to the n
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who's Hamming weight--
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so Hamming weight here
is just the number
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of 1's or it's a number
of non-zero elements--
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number of 1's in
x or, in general,
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the number of non-zero
elements among the coordinates
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of x, so the number of
non-zero coordinates.
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And I want the Hamming weight
to be less than this quantity
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here.
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So visually, what
this looks like is
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I'm thinking of the
Hamming cube placed
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so that the all-zero
vector is here,
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the all-ones vector is up there.
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And then it's sorted
by Hamming weight.
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So this is called
a Boolean lattice.
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And I'm looking at
all the elements, A,
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which are within a Hamming
ball of the 0 vector.
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So this is the set.
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It's not too hard to
calculate the size of the set,
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because I'm taking everything
with Hamming weight
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less than this quantity over
here by central limit theorem.
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The number of elements
in the set is of the form
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a constant fraction
of the entire space,
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where alpha is some
constant if c is a constant.
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So it has the desired size.
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But also, A added
to itself consists
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of points in the Boolean
cube whose Hamming weight
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is at most n minus c root n.
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And I claim that
this sumset does not
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contain any subspace of
dimension larger than n minus c
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root n.
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So this is the final claim.
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It's something that's, again,
one of these linear algebraic
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exercises that we've actually
seen earlier when we discussed
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the proof of a cap set, right,
so the polynomial method
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proof of cap set.
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If you have a
subspace of dimension
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greater than some
quantity, then you
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should be able to find a
vector in that subspace whose
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support has size at
least the dimension.
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OK.
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So you see, in
particular, we do not
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have any bounded codimensional
subspaces in this A plus A.
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So even though the
philosophy is roughly right
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that if you start with a set
A and you add it to itself,
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it smooths out
the set, it should
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contain-- we expect it to
contain some large structure.
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That's not quite true.
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But what turns out to be true,
and this is the first result
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that I will show today,
is that if you add A
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to itself a few more
times, that, indeed,
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you can get large subspaces.
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And this is an important step
in a proof of Freiman's theorem.
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And this step is known
as Bogolyubov's lemma.
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So Bogolyubov's lemma, in
the case of F2 to the n,
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says that if you
have a subset A of F2
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to the n of fraction alpha of
the space, then 2A minus 2A
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contains a bounded codimensional
subspace, so a very large
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subspace.
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so 2A minus 2A contains
a subspace of codimension
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less than 1 over alpha squared.
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Here, I write 2A minus 2A
even though we're in F2.
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So this is the same as 4A.
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But in general-- and you'll
see later on when we do it
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in integers--
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2A minus 2A is the right
expression to look at.
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So 2A minus 2A, something
that works in every group.
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But for F2 to the n,
it's the same as 4A.
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So the main philosophy here
is that adding is smoothing.
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You start with a large
subset of F2 to the n.
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It's large.
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Does it contain
large structures?
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Not necessarily.
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But you add it to itself, and
it smooths out the picture.
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So it has a rough spot.
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It smooths it out.
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And if you keep
adding A to itself,
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it smooths it out even further.
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You add it to
itself enough times,
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and then it will contain
a large structure
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and just from the
size of A alone.
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And there is a very
similar idea, which
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comes up all over the
place in analysis,
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is that convolutions
are smoothing.
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So you start with some function
that might be very rough.
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If you convolve it
with itself and if you
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do it many more times,
you get something
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that is much smoother.
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And in fact, adding
and convolutions
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are almost the same things.
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And I'll explain
that in a second.
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So this is an important idea
to take away from all of this.
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So when we do these free
analytic calculations--
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so there will be some free
analytic calculations--
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the first time you see
them, they might just
00:11:56.213 --> 00:11:57.720
seem like calculations.
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So you push the symbols around,
and you get some inequalities.
00:12:00.630 --> 00:12:01.640
You get some answers.
00:12:01.640 --> 00:12:04.203
But that's no way to
learn the subject.
00:12:04.203 --> 00:12:05.620
So you need to
figure out, what is
00:12:05.620 --> 00:12:07.690
the intuition behind each step?
00:12:07.690 --> 00:12:10.280
Because when you need
to work on it yourself,
00:12:10.280 --> 00:12:13.430
you're not just guessing the
right symbols to put down.
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You have to understand
the intuition, like why,
00:12:16.760 --> 00:12:20.020
intuitively, each inequality
should be expected to hold?
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And this is an important
idea that adding is smoothing
00:12:22.550 --> 00:12:25.128
and convolution is smoothing.
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All right.
00:12:26.000 --> 00:12:28.415
So let me remind you
about convolutions.
00:12:34.000 --> 00:12:36.680
So recall, in a
general abelian group,
00:12:36.680 --> 00:12:39.930
if I have two functions,
f and g, on the group--
00:12:39.930 --> 00:12:43.100
so complex value functions--
then the convolution
00:12:43.100 --> 00:12:45.990
is given by the
following formula.
00:12:49.520 --> 00:12:53.550
So that's the convolution.
00:12:53.550 --> 00:12:56.130
And it behaves very
well with respect
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to the Fourier transform.
00:12:58.350 --> 00:13:04.020
The Fourier transform
turns convolutions
00:13:04.020 --> 00:13:05.910
into multiplications.
00:13:05.910 --> 00:13:12.600
So this means, point
wise, I have that.
00:13:12.600 --> 00:13:17.240
Convolutions also relate to
some sets, because if I--
00:13:17.240 --> 00:13:19.260
and this is the
interpretation of convolutions
00:13:19.260 --> 00:13:21.420
that I want you to keep
in mind for the purpose
00:13:21.420 --> 00:13:24.060
of additive combinatorics.
00:13:24.060 --> 00:13:26.790
If you have two
sets, A and B, then
00:13:26.790 --> 00:13:29.670
look at the convolution
of their indicators.
00:13:33.090 --> 00:13:35.050
It has an interpretation.
00:13:35.050 --> 00:13:37.590
So if you read out
what this value says,
00:13:37.590 --> 00:13:41.670
then what this comes
out to is 1 divided
00:13:41.670 --> 00:13:48.450
by the size of the group over
the number of pairs in A and B
00:13:48.450 --> 00:13:52.860
such that their sum is x.
00:13:52.860 --> 00:13:55.960
So up to normalization,
convolution records
00:13:55.960 --> 00:13:58.540
some sets with multiplicities.
00:13:58.540 --> 00:14:01.380
So the convolution
tells you how many ways
00:14:01.380 --> 00:14:05.130
are there to express x in
terms of a sum of one element
00:14:05.130 --> 00:14:07.470
from A and another
element from B.
00:14:07.470 --> 00:14:09.810
And in particular,
this function here
00:14:09.810 --> 00:14:15.930
is supported on the
sumset A plus B.
00:14:15.930 --> 00:14:19.440
So this is the way that
convolutions and sumsets
00:14:19.440 --> 00:14:21.300
are intimately
related to each other.
00:14:24.100 --> 00:14:26.550
So that's proof
Bogolyubov's lemma.
00:14:33.930 --> 00:14:40.050
We're going to be looking at
this sumset, which is related
00:14:40.050 --> 00:14:42.160
to the following convolution.
00:14:42.160 --> 00:14:51.910
So let f be this convolution
of indicators, AA minus A minus
00:14:51.910 --> 00:14:53.285
A. Of course, in
F2 to the n, you
00:14:53.285 --> 00:14:55.035
don't need to worry
about the minus signs.
00:14:55.035 --> 00:14:57.710
But I'll keep them there
for future reference.
00:14:57.710 --> 00:15:03.760
Here, by what we said
earlier, the support of f
00:15:03.760 --> 00:15:06.130
is 2A minus 2A.
00:15:09.480 --> 00:15:12.960
It's not too hard to evaluate
the Fourier transform of f,
00:15:12.960 --> 00:15:17.050
because Fourier transform plays
very well with convolutions.
00:15:17.050 --> 00:15:21.390
So in this case, it is the
Fourier transform of A squared,
00:15:21.390 --> 00:15:24.380
Fourier transform
of minus A squared.
00:15:24.380 --> 00:15:27.360
The Fourier transform of minus
A, if you look at the formula,
00:15:27.360 --> 00:15:28.540
is the conjugate.
00:15:28.540 --> 00:15:31.440
It's a complex conjugate of
the Fourier transform of A.
00:15:31.440 --> 00:15:36.350
So again, in F2 to the n,
they're actually the same.
00:15:36.350 --> 00:15:41.610
But in general, it's
the complex conjugate.
00:15:41.610 --> 00:15:44.130
So we always have
this formula here.
00:15:48.150 --> 00:15:50.910
So we always have that.
00:15:50.910 --> 00:15:56.410
So by the Fourier
inversion formula,
00:15:56.410 --> 00:16:02.082
we can write f in terms
of its Fourier transform.
00:16:07.164 --> 00:16:08.265
Here, I'm using that.
00:16:08.265 --> 00:16:09.140
We're now using that.
00:16:09.140 --> 00:16:11.780
We're in F2, so that's
what the inverse Fourier
00:16:11.780 --> 00:16:14.090
transform looks like.
00:16:14.090 --> 00:16:22.780
And so we have the
following formula
00:16:22.780 --> 00:16:30.130
for the value of f in terms
of the Fourier transform
00:16:30.130 --> 00:16:33.050
of the original set.
00:16:35.820 --> 00:16:36.510
OK.
00:16:36.510 --> 00:16:43.650
We want to show that f, whose
support is the 2A minus 2A
00:16:43.650 --> 00:16:44.770
we're interested in--
00:16:44.770 --> 00:16:47.340
we want to show the
support of f contains
00:16:47.340 --> 00:16:52.390
a large subspace, a small,
codimensional subspace.
00:16:52.390 --> 00:17:02.250
So observe that if f has
positive value, then--
00:17:07.089 --> 00:17:12.010
if f of x is positive,
then x lies in its support.
00:17:12.010 --> 00:17:15.760
So we just want to
find a large subspace
00:17:15.760 --> 00:17:20.290
on which f is positive.
00:17:20.290 --> 00:17:24.250
But we can choose our
subspace by looking
00:17:24.250 --> 00:17:29.120
at Fourier coefficients
according to their size.
00:17:29.120 --> 00:17:41.640
So what we can do is let R
be the set of, essentially,
00:17:41.640 --> 00:17:49.900
Fourier characters whose
corresponding value
00:17:49.900 --> 00:17:52.360
in the Fourier
transform is large.
00:17:55.130 --> 00:17:57.100
So it's at least
alpha to the 3/2.
00:17:57.100 --> 00:18:01.000
And that value
will come up later.
00:18:01.000 --> 00:18:03.480
So let's look at
this R. And what
00:18:03.480 --> 00:18:05.550
we're going to do is
we're going to look
00:18:05.550 --> 00:18:08.040
at the orthogonal
complement of R
00:18:08.040 --> 00:18:16.830
and show that f is positive on
the orthogonal complement of R.
00:18:16.830 --> 00:18:19.590
First, R is not too large.
00:18:19.590 --> 00:18:27.880
The size of R is, I claim,
less than 1 over alpha squared.
00:18:27.880 --> 00:18:28.856
Why's that?
00:18:33.233 --> 00:18:34.900
So this is an important
trick that we've
00:18:34.900 --> 00:18:36.280
seen a few times before.
00:18:38.810 --> 00:18:42.950
The number of large Fourier
coefficients cannot be too
00:18:42.950 --> 00:18:51.320
large, because they're Parseval,
which tells us that the sum
00:18:51.320 --> 00:18:59.550
of the squares of the Fourier
coefficients is equal to the L2
00:18:59.550 --> 00:19:05.680
norm of the original
function, which, in this case,
00:19:05.680 --> 00:19:09.480
is just the density
of A. So that's alpha.
00:19:09.480 --> 00:19:13.450
So just looking at that, the
number of large terms cannot be
00:19:13.450 --> 00:19:13.950
too many.
00:19:18.650 --> 00:19:20.590
OK.
00:19:20.590 --> 00:19:25.810
So we have this small set, R,
on which f has large Fourier
00:19:25.810 --> 00:19:27.150
transform values.
00:19:31.390 --> 00:19:35.730
Now, let's look at f of x.
00:19:35.730 --> 00:19:36.800
So let's look at f of x.
00:19:36.800 --> 00:19:42.010
We want to find out, when
can we control f of x
00:19:42.010 --> 00:19:45.850
to make sure it is positive?
00:19:45.850 --> 00:19:50.020
Well, for the values of r--
00:19:50.020 --> 00:19:58.490
little r-- not in big R or
0, we see that the Fourier
00:19:58.490 --> 00:20:01.770
transform--
00:20:01.770 --> 00:20:04.890
we would like to upper bound
this quantity here so that this
00:20:04.890 --> 00:20:05.760
is negligible.
00:20:05.760 --> 00:20:07.195
This is a small term.
00:20:07.195 --> 00:20:08.820
Again, this is a
computation that we've
00:20:08.820 --> 00:20:11.950
seen several times
earlier in this course.
00:20:11.950 --> 00:20:13.840
All of these terms are small.
00:20:13.840 --> 00:20:16.750
So I want to show that
the whole sum is small.
00:20:16.750 --> 00:20:20.050
I don't want to bound each term
individually and then sum up
00:20:20.050 --> 00:20:21.670
all the possible contributions.
00:20:21.670 --> 00:20:22.780
That will be too big.
00:20:22.780 --> 00:20:25.240
But we've seen this trick
before where we just take out
00:20:25.240 --> 00:20:27.725
some subset of the factors.
00:20:27.725 --> 00:20:31.150
So in particular, I'll
take out two of the factors
00:20:31.150 --> 00:20:45.495
and get alpha cubed upper bound
plus the remaining factors.
00:20:52.260 --> 00:20:58.900
And once again, use Parseval
on this very last sum,
00:20:58.900 --> 00:21:02.700
keeping in mind that
I'm throwing away
00:21:02.700 --> 00:21:05.670
some of the r's, including 0.
00:21:05.670 --> 00:21:08.509
So it will be a
strict inequality.
00:21:14.365 --> 00:21:14.865
OK.
00:21:14.865 --> 00:21:15.365
Yeah.
00:21:15.365 --> 00:21:17.910
So this step should be
reminiscent of very similar
00:21:17.910 --> 00:21:20.580
computations that we did in
the proof of Roth's theorem.
00:21:23.510 --> 00:21:34.640
So if x lies in the orthogonal
complement of uppercase R, then
00:21:34.640 --> 00:21:35.840
f of x--
00:21:35.840 --> 00:21:40.040
well, let's evaluate f of x from
the Fourier inversion formula.
00:21:40.040 --> 00:21:47.540
We have this.
00:21:52.600 --> 00:22:06.815
So I can now split the sum as
the 0-th term, the large terms.
00:22:11.337 --> 00:22:14.660
Now, you see, for
the large terms,
00:22:14.660 --> 00:22:17.750
because we're in the
orthogonal complement of A,
00:22:17.750 --> 00:22:20.170
I can make sure that they all
come with a positive sign.
00:22:24.510 --> 00:22:26.450
And finally, the small terms.
00:22:42.930 --> 00:22:49.080
And you see that the main
term is alpha to the 4.
00:22:51.780 --> 00:22:56.980
This term is always
non-negative.
00:22:56.980 --> 00:22:59.860
And the error terms,
the small terms,
00:22:59.860 --> 00:23:05.960
are strictly less than alpha
to the 4th in magnitude.
00:23:08.780 --> 00:23:14.070
So as a result, this
whole sum is positive.
00:23:14.070 --> 00:23:14.570
Yeah.
00:23:14.570 --> 00:23:17.233
AUDIENCE: These 1's are
also 1 sub A's, right?
00:23:17.233 --> 00:23:18.150
YUFEI ZHAO: Thank you.
00:23:18.150 --> 00:23:20.658
The 1's are 1 sub A's.
00:23:20.658 --> 00:23:21.158
Yeah.
00:23:24.470 --> 00:23:26.130
So this is the very
similar philosophy
00:23:26.130 --> 00:23:28.920
to when we proved
Roth's theorem.
00:23:28.920 --> 00:23:33.670
We look at a sum like this,
so some trigonometric series,
00:23:33.670 --> 00:23:35.220
some Fourier series.
00:23:35.220 --> 00:23:38.460
And we decompose it
into several terms
00:23:38.460 --> 00:23:43.500
based on how large their
Fourier coefficients are.
00:23:43.500 --> 00:23:46.840
We can control the small ones
using what essentially amounts
00:23:46.840 --> 00:23:54.070
to a counting lemma and show
that the small ones cannot ever
00:23:54.070 --> 00:23:59.140
annihilate the large,
dominant terms.
00:23:59.140 --> 00:24:03.540
So as a result, f
of x is positive
00:24:03.540 --> 00:24:06.240
on the orthogonal
complement of R.
00:24:06.240 --> 00:24:14.600
So thus R lies in
the support of f,
00:24:14.600 --> 00:24:17.600
which is equal to 2A minus 2A.
00:24:21.300 --> 00:24:34.342
And furthermore, the
codimension of R is at most--
00:24:34.342 --> 00:24:36.050
so it could be some
linear dependencies--
00:24:36.050 --> 00:24:39.990
is at most the size of R,
which is strictly less than 1
00:24:39.990 --> 00:24:40.850
over alpha squared.
00:24:45.020 --> 00:24:47.250
And that proves
Bogolyubov's lemma.
00:24:47.250 --> 00:24:50.170
So if you have a large
subset of F2 to the n,
00:24:50.170 --> 00:24:52.980
you add it to
itself enough times
00:24:52.980 --> 00:24:55.030
so that it's a
smoothing operation.
00:24:55.030 --> 00:24:57.790
And then eventually, you
must find a large structure.
00:24:57.790 --> 00:25:00.340
And we only start by
assuming the size of it.
00:25:00.340 --> 00:25:01.930
If it's just large
enough, then we
00:25:01.930 --> 00:25:05.350
can find a large structure
within this iterated sumset.
00:25:07.910 --> 00:25:08.954
Any questions?
00:25:12.720 --> 00:25:13.220
Yeah?
00:25:13.220 --> 00:25:17.462
AUDIENCE: Isn't R in
support of R [INAUDIBLE]??
00:25:20.240 --> 00:25:21.764
YUFEI ZHAO: Sorry, come again?
00:25:21.764 --> 00:25:24.740
AUDIENCE: You got that the
orthogonal complement of R--
00:25:24.740 --> 00:25:25.490
YUFEI ZHAO: Sorry.
00:25:25.490 --> 00:25:27.817
The orthogonal complement
of R is in the support.
00:25:27.817 --> 00:25:28.571
Yeah.
00:25:28.571 --> 00:25:31.560
So R lives in the
character space.
00:25:34.530 --> 00:25:35.430
OK, great.
00:25:35.430 --> 00:25:37.740
So this is the proof
of Bogolyubov's lemma
00:25:37.740 --> 00:25:39.960
in the finite field
setting, working
00:25:39.960 --> 00:25:43.590
in F2 to the n, which is fine.
00:25:43.590 --> 00:25:47.310
It's a useful setting as a
playground for us to work in.
00:25:47.310 --> 00:25:48.990
But ultimately, we
want to understand
00:25:48.990 --> 00:25:50.830
what happens in the integers.
00:25:50.830 --> 00:25:52.860
So if you look at where
we left off last time,
00:25:52.860 --> 00:25:56.610
we started in the
cyclic group, Z mod n.
00:25:56.610 --> 00:26:01.140
So we would like to know how
to formulate a similar result
00:26:01.140 --> 00:26:07.980
but in the cyclic group where
there are no more subspaces.
00:26:07.980 --> 00:26:09.822
We encountered a
similar situation,
00:26:09.822 --> 00:26:11.280
although we didn't
go into it, when
00:26:11.280 --> 00:26:12.990
we discussed Roth's theorem.
00:26:12.990 --> 00:26:15.150
In the first proof
of Roth's theorem
00:26:15.150 --> 00:26:18.960
that we showed, in the
first Fourier analytic proof
00:26:18.960 --> 00:26:20.910
in the finite field
setting, the proof
00:26:20.910 --> 00:26:26.570
won by restricting to
subspaces, to hyperplanes.
00:26:26.570 --> 00:26:30.530
And then we keep on iterating
by restricting to hyperplanes.
00:26:30.530 --> 00:26:32.230
So you can stay in subspaces.
00:26:32.230 --> 00:26:37.110
And the finite field setting
has lots of subspaces.
00:26:37.110 --> 00:26:40.640
And we said that to get that
proof to work in the integers,
00:26:40.640 --> 00:26:43.400
we had to do
something different.
00:26:43.400 --> 00:26:46.710
And we did something by
restricting to intervals.
00:26:46.710 --> 00:26:48.460
But I also mentioned
that, somehow, that's
00:26:48.460 --> 00:26:51.100
not the natural
analog of subspaces.
00:26:51.100 --> 00:26:54.880
The natural analog of subspaces
is something called a Bohr set.
00:26:54.880 --> 00:26:57.480
And so I want to explore
this idea further now.
00:27:00.650 --> 00:27:05.160
So the natural analog
of subspaces in Z mod n
00:27:05.160 --> 00:27:06.750
are these objects
called Bohr sets.
00:27:09.820 --> 00:27:11.980
And they're defined as follows.
00:27:11.980 --> 00:27:19.840
So suppose you are given
some R, a subset of Z mod n.
00:27:19.840 --> 00:27:24.310
We define a Bohr set,
denoted like this, so
00:27:24.310 --> 00:27:34.220
Bohr of R and epsilon, to
be the subset of Z mod n,
00:27:34.220 --> 00:27:40.150
so including elements
x, such that rx
00:27:40.150 --> 00:27:44.330
is pretty close to
a multiple of n.
00:27:44.330 --> 00:27:48.220
So here, we're looking
at the R mod Z norm.
00:27:48.220 --> 00:27:52.200
So this is the distance to
the closest integer such
00:27:52.200 --> 00:27:59.120
that this fraction is very close
to an integer for all little r
00:27:59.120 --> 00:28:08.872
and big R. You see, this
is the analog of subspaces,
00:28:08.872 --> 00:28:11.080
because in the finite field
setting, the finite field
00:28:11.080 --> 00:28:16.010
vector space, even if I
set epsilon to equal to 0
00:28:16.010 --> 00:28:18.800
and turn this into
an inner product,
00:28:18.800 --> 00:28:23.360
then Bohr sets are exactly
subspaces-- namely,
00:28:23.360 --> 00:28:27.730
the orthogonal
complement of the set R.
00:28:27.730 --> 00:28:31.600
But now we're in the integers,
where you don't have exact 0.
00:28:31.600 --> 00:28:35.320
But I just want that quantity,
that norm, to be small enough.
00:28:40.020 --> 00:28:41.460
So let me give you some names.
00:28:41.460 --> 00:28:43.800
So given the Bohr set,
which, technically
00:28:43.800 --> 00:28:45.720
speaking, is more
than just the set
00:28:45.720 --> 00:28:49.950
itself but also includes the
information of R and epsilon--
00:28:49.950 --> 00:28:52.470
so it's the entire data
written on the board--
00:28:52.470 --> 00:28:58.530
we call the size of the R
the dimension of the Bohr set
00:28:58.530 --> 00:28:59.880
and epsilon, the width.
00:29:04.690 --> 00:29:11.230
Bogolyubov's lemma for Z mod n
now takes the following form.
00:29:20.960 --> 00:29:25.880
If you start with a subset A of
Z mod n, and all I need to know
00:29:25.880 --> 00:29:32.450
is that A is a constant
fraction of the cyclic group,
00:29:32.450 --> 00:29:35.870
then the iterated
sumset 2A minus 2A
00:29:35.870 --> 00:29:51.250
contains some Bohr
set Bohr R of 1/4
00:29:51.250 --> 00:29:56.740
with the size of R less
than 1 over alpha squared.
00:30:01.400 --> 00:30:03.970
So earlier, we said that if
you have a large subset of F2
00:30:03.970 --> 00:30:08.990
to the n, then 2A minus 2A
contains a large subspace.
00:30:08.990 --> 00:30:11.930
And now we say that
if A is a large subset
00:30:11.930 --> 00:30:14.450
of the cyclic group,
then 2A minus 2A
00:30:14.450 --> 00:30:19.130
contains a large Bohr
set of small dimension.
00:30:19.130 --> 00:30:22.790
And so this terminology
may be slightly confusing.
00:30:22.790 --> 00:30:27.090
The dimension corresponds
to codimension previously.
00:30:27.090 --> 00:30:33.020
So if you do this
translation, this dimension--
00:30:33.020 --> 00:30:35.078
I mean, if R were a set
of independent vectors
00:30:35.078 --> 00:30:36.620
and you have 2 to
the n, then that'll
00:30:36.620 --> 00:30:39.290
be the codimension of the
corresponding subspace.
00:30:39.290 --> 00:30:43.770
But this is the terminology
that we're stuck with.
00:30:43.770 --> 00:30:44.270
OK.
00:30:44.270 --> 00:30:46.060
Any questions about
the statement?
00:30:50.770 --> 00:30:53.590
You see, even the bounds are
exactly the same, 1 over alpha
00:30:53.590 --> 00:30:54.430
squared.
00:30:54.430 --> 00:30:56.320
And I mean, the
proof is going to be
00:30:56.320 --> 00:30:57.970
pretty much exactly
the same once you
00:30:57.970 --> 00:31:01.247
make the correct
notational modifications.
00:31:01.247 --> 00:31:02.330
So we're going to do that.
00:31:02.330 --> 00:31:05.230
So I'm going to write on
top of this earlier proof
00:31:05.230 --> 00:31:08.080
and show you what are the
notational modifications so
00:31:08.080 --> 00:31:12.640
that you can get exactly
the same result here
00:31:12.640 --> 00:31:14.760
but with Bohr sets
instead of a subspace.
00:31:17.840 --> 00:31:20.690
The thing to keep in mind
is that we have a somewhat
00:31:20.690 --> 00:31:23.210
different Fourier transform.
00:31:23.210 --> 00:31:28.580
So let me now use
different colored chalk.
00:31:28.580 --> 00:31:40.870
So the Fourier transform of
a function f from Z mod n,
00:31:40.870 --> 00:31:46.330
so a complex value,
is a function
00:31:46.330 --> 00:31:55.570
also on Z mod n defined by f
hat of r equal to expectation
00:31:55.570 --> 00:32:00.100
over x in Z mod n of
f of x times omega
00:32:00.100 --> 00:32:07.690
to the minus rx, where omega
is a primitive n root of unity.
00:32:17.950 --> 00:32:20.010
And you also had the
Fourier inversion formula.
00:32:20.010 --> 00:32:21.500
It's what you expect.
00:32:21.500 --> 00:32:23.735
I won't bother writing it down.
00:32:23.735 --> 00:32:24.860
So we go back to the proof.
00:32:24.860 --> 00:32:27.550
And pretty much everything
will read exactly the same.
00:32:27.550 --> 00:32:28.920
So f is still the same f.
00:32:32.180 --> 00:32:36.866
And the Fourier transform
has the same property.
00:32:36.866 --> 00:32:40.880
So all of these nice properties
of the Fourier transform hold.
00:32:40.880 --> 00:32:43.340
For inversion, it's
basically the same
00:32:43.340 --> 00:32:48.210
except that the formula
is slightly different.
00:32:48.210 --> 00:32:50.960
So instead of minus
1 to the r dot
00:32:50.960 --> 00:32:54.700
x, what we have now
is omega to the rx.
00:32:58.310 --> 00:33:01.990
So here, we have
omega to the rx.
00:33:05.530 --> 00:33:06.193
OK.
00:33:06.193 --> 00:33:08.660
Great.
00:33:08.660 --> 00:33:12.620
The next part is the
same, where we define r.
00:33:12.620 --> 00:33:21.680
So now we define r to consist
of elements of z mod n, whose
00:33:21.680 --> 00:33:24.190
Fourier transform is large.
00:33:24.190 --> 00:33:25.170
I can take out 0.
00:33:36.555 --> 00:33:39.433
OK.
00:33:39.433 --> 00:33:40.600
This part is still the same.
00:33:40.600 --> 00:33:42.540
It's the same calculation.
00:33:42.540 --> 00:33:44.340
Now, it's the very
last part that needs
00:33:44.340 --> 00:33:46.320
to be just slightly changed.
00:33:49.890 --> 00:33:53.030
Where does the 1/4 come in?
00:33:53.030 --> 00:33:54.990
So where does this come in?
00:33:54.990 --> 00:34:06.120
So observe that if x is in
the Bohr set with width 1/4,
00:34:06.120 --> 00:34:16.830
then rx divided by n is--
00:34:16.830 --> 00:34:20.860
OK, so by definition,
all of these fractions
00:34:20.860 --> 00:34:24.070
are within the
1/4 of an integer.
00:34:24.070 --> 00:34:29.320
And if you think about what
happens on the unit circle,
00:34:29.320 --> 00:34:33.250
if you are within
1/4 of the integer,
00:34:33.250 --> 00:34:36.550
then that means the
corresponding place on the unit
00:34:36.550 --> 00:34:44.900
circle is on the
left half circle.
00:34:44.900 --> 00:34:54.630
So in particular, the cosine
of 2rx over n is non-negative.
00:34:54.630 --> 00:35:00.110
So it has non-negative
real part.
00:35:00.110 --> 00:35:02.980
So now we go back to
this part of the proof,
00:35:02.980 --> 00:35:07.180
where we're applying Fourier
inversion formula to f of x.
00:35:07.180 --> 00:35:09.430
So we had the Fourier
inversion formula up there.
00:35:09.430 --> 00:35:16.160
But because f of x is real,
it's really the cosine
00:35:16.160 --> 00:35:18.660
that should come in play.
00:35:21.795 --> 00:35:23.580
It should be a cosine.
00:35:23.580 --> 00:35:31.320
And now, for the next step,
we have no negative sign here,
00:35:31.320 --> 00:35:34.510
because this step--
00:35:34.510 --> 00:35:40.420
OK, let me just cross
out this step over there.
00:35:40.420 --> 00:35:43.000
All of these terms,
the terms that
00:35:43.000 --> 00:35:45.190
correspond to
little r and big R,
00:35:45.190 --> 00:35:49.380
they have non-negative
contribution.
00:35:53.080 --> 00:35:55.840
Whatever the contributions
here, it's non-negative.
00:35:55.840 --> 00:35:59.480
So I cross out this term.
00:35:59.480 --> 00:36:01.580
All I'm left with
is the main term,
00:36:01.580 --> 00:36:05.310
corresponding to the
density, and the error term,
00:36:05.310 --> 00:36:08.540
so to speak, the minor terms,
which is less than alpha
00:36:08.540 --> 00:36:12.315
to the 4th in absolute value.
00:36:12.315 --> 00:36:14.190
OK.
00:36:14.190 --> 00:36:17.883
So it's positive.
00:36:17.883 --> 00:36:19.050
So basically the same proof.
00:36:19.050 --> 00:36:21.210
Once you make the
appropriate modifications,
00:36:21.210 --> 00:36:23.140
it's the same proof in Z mod n.
00:36:28.320 --> 00:36:29.460
OK, great.
00:36:29.460 --> 00:36:33.602
So this concludes our discussion
of Bogolyubov's lemma.
00:36:33.602 --> 00:36:35.550
So it says that--
00:36:35.550 --> 00:36:37.540
OK, so continuing
our previous thread,
00:36:37.540 --> 00:36:42.670
we start with a subset of z
mod n of constant proportion.
00:36:42.670 --> 00:36:49.820
Then 2A minus 2A necessarily
contains a large Bohr set.
00:36:49.820 --> 00:36:53.285
And the next thing I want to do
is to start with this Bohr set.
00:36:53.285 --> 00:36:54.910
So that's the definition
of a Bohr set.
00:36:54.910 --> 00:36:56.100
But what does it look like?
00:36:56.100 --> 00:36:59.887
So it's a bit hard to imagine.
00:36:59.887 --> 00:37:00.970
So what does it look like?
00:37:00.970 --> 00:37:04.390
In the finite field setting,
we know it's a subspace.
00:37:04.390 --> 00:37:07.510
But in the Z mod n
setting, right now, it's
00:37:07.510 --> 00:37:09.550
just some subset of Z mod n.
00:37:09.550 --> 00:37:12.520
OK, so in the next
step, we want to extract
00:37:12.520 --> 00:37:18.610
some geometric structure
from this Bohr set.
00:37:18.610 --> 00:37:20.830
So we're going to
show that this Bohr
00:37:20.830 --> 00:37:24.940
set will contain a large,
generalized arithmetic
00:37:24.940 --> 00:37:25.802
progression.
00:37:28.460 --> 00:37:30.790
So you asked something
earlier about--
00:37:30.790 --> 00:37:34.510
something seems a bit fishy
about the general strategy.
00:37:34.510 --> 00:37:38.500
Seems like our
goal for proving--
00:37:38.500 --> 00:37:40.630
we want to prove
Freiman's theorem, which
00:37:40.630 --> 00:37:43.640
says that the
conclusion is that A
00:37:43.640 --> 00:37:50.890
is contained in some GAP,
some fairly compact additive
00:37:50.890 --> 00:37:52.770
structure.
00:37:52.770 --> 00:37:54.850
And we're already
losing quite a bit.
00:37:54.850 --> 00:37:58.240
So we pass down
to 1/8 of A. So it
00:37:58.240 --> 00:38:02.110
seems like even if
you contain the rest,
00:38:02.110 --> 00:38:05.230
even if you can contain this
fraction, this large fraction
00:38:05.230 --> 00:38:10.270
of A, what are you going
to do about the rest of A?
00:38:10.270 --> 00:38:12.440
That's an unanswered question.
00:38:12.440 --> 00:38:14.440
A second unanswered
question-- so right now,
00:38:14.440 --> 00:38:16.570
what I've told you,
the strategy is
00:38:16.570 --> 00:38:25.280
we're going to find a large
GAP inside 2A minus 2A,
00:38:25.280 --> 00:38:28.710
which is not quite the
thing that we want to do.
00:38:28.710 --> 00:38:32.880
We want to contain
A in a small GAP.
00:38:32.880 --> 00:38:35.050
But at least it's
some progress, right?
00:38:35.050 --> 00:38:37.140
It's some progress to
find some structure.
00:38:37.140 --> 00:38:38.880
I mean, the name of
the game is to try
00:38:38.880 --> 00:38:40.890
to find additive structure.
00:38:40.890 --> 00:38:43.590
So in the theme of
this whole semester
00:38:43.590 --> 00:38:46.770
course is trying to understand
the dichotomy between structure
00:38:46.770 --> 00:38:48.050
and pseudorandomness.
00:38:48.050 --> 00:38:50.960
And when you have structure,
let's use that structure.
00:38:50.960 --> 00:38:53.820
See if you can boost
that structure.
00:38:53.820 --> 00:38:57.150
So there will be an
additional argument, which
00:38:57.150 --> 00:38:59.880
I will show you at the
beginning of next lecture
00:38:59.880 --> 00:39:04.380
at the conclusion of the proof
of Freiman's theorem, which
00:39:04.380 --> 00:39:08.460
will allow you to start with the
structure on a small part of A,
00:39:08.460 --> 00:39:10.950
but not too small-- it's
a constant fraction of A--
00:39:10.950 --> 00:39:15.516
and pass it up to
the whole of A.
00:39:15.516 --> 00:39:17.670
And we've actually
already seen a tool
00:39:17.670 --> 00:39:20.990
that allows us to do that.
00:39:20.990 --> 00:39:26.610
So I want to cover all
of A. So last time, we
00:39:26.610 --> 00:39:28.530
did something called
the covering lemma,
00:39:28.530 --> 00:39:30.600
Ruzsa covering
lemma, that tells us
00:39:30.600 --> 00:39:33.585
that if you have some
nice control on A
00:39:33.585 --> 00:39:38.160
and you can cover some
part of A very well,
00:39:38.160 --> 00:39:42.110
then I can cover the
entirety of A very well.
00:39:42.110 --> 00:39:45.460
So those tools
will come in hand.
00:39:45.460 --> 00:39:47.100
I mean, so similar
to actually how we
00:39:47.100 --> 00:39:50.357
proved Freiman's theorem in
groups with bounded exponent.
00:39:50.357 --> 00:39:51.940
And so we're going
to use the covering
00:39:51.940 --> 00:39:54.540
lemma to conclude the theorem.
00:39:57.290 --> 00:39:58.790
But now I want to
get into the issue
00:39:58.790 --> 00:40:00.140
of the geometry of numbers.
00:40:08.900 --> 00:40:09.570
OK.
00:40:09.570 --> 00:40:11.580
I want to tell you
some necessary tools
00:40:11.580 --> 00:40:19.632
that we'll need to find a
large GAP inside 2A minus 2A.
00:40:19.632 --> 00:40:23.750
Now, it will seem like
a bit of a digression,
00:40:23.750 --> 00:40:26.050
but we'll come back into
additive combinatorics
00:40:26.050 --> 00:40:27.070
in a bit.
00:40:27.070 --> 00:40:30.155
So the geometry of numbers
concerns the study of lattices.
00:40:32.910 --> 00:40:37.200
So it concerns the study of
lattices and convex bodies.
00:40:41.370 --> 00:40:45.710
So this is a really important
area of mathematics, especially
00:40:45.710 --> 00:40:49.220
about a century ago with
mathematicians like Minkowski
00:40:49.220 --> 00:40:51.340
playing foundational
roles in the subject.
00:40:51.340 --> 00:40:54.050
So number theorists were
very interested in trying
00:40:54.050 --> 00:40:58.100
to understand how
lattices behave.
00:40:58.100 --> 00:41:01.340
So I'll tell you some
very classical results
00:41:01.340 --> 00:41:04.580
that we'll use for
proving Freiman's theorem.
00:41:08.140 --> 00:41:10.270
So first, what is a lattice?
00:41:10.270 --> 00:41:16.870
So let me give you the following
definition of a lattice in R
00:41:16.870 --> 00:41:18.390
to the d.
00:41:18.390 --> 00:41:26.910
It's a structure on a group,
if you will, as an integer
00:41:26.910 --> 00:41:31.270
span of d independent vectors.
00:41:37.460 --> 00:41:43.080
So I start with v1
through vd vectors
00:41:43.080 --> 00:41:45.890
that are linearly independent.
00:41:45.890 --> 00:41:47.730
And I look at
their integer span.
00:41:47.730 --> 00:41:50.710
I think this is best
explained with a picture.
00:41:50.710 --> 00:41:55.790
So if I have a bunch of--
00:41:58.900 --> 00:42:00.760
so here, I'm drawing
a picture in R2.
00:42:06.350 --> 00:42:09.080
And this picture extends
in all directions.
00:42:09.080 --> 00:42:15.460
If I start with two vectors, v1
and v2, linearly independent,
00:42:15.460 --> 00:42:20.080
and look at their integer
span, so that's a lattice.
00:42:20.080 --> 00:42:21.703
So that's what a lattice is.
00:42:21.703 --> 00:42:23.870
You can come up with all
sorts of fancy definitions,
00:42:23.870 --> 00:42:28.250
like a discrete
subgroup of R to the n.
00:42:28.250 --> 00:42:29.540
But this is what it is.
00:42:32.740 --> 00:42:36.670
So just to emphasize this
definition for a bit--
00:42:36.670 --> 00:42:39.400
and also, one more
definition that we'll need
00:42:39.400 --> 00:42:43.100
is the determinant of a lattice.
00:42:43.100 --> 00:42:46.530
So what's the
determinant of a lattice?
00:42:46.530 --> 00:42:51.120
One way to define it is you
look at these v's, and you
00:42:51.120 --> 00:42:59.630
construct a matrix with
the v's as columns.
00:42:59.630 --> 00:43:05.070
And you evaluate the absolute
value of this determinant.
00:43:05.070 --> 00:43:11.720
More visually, the determinant
of a lattice is also equal
00:43:11.720 --> 00:43:24.100
to the volume of its
fundamental parallelepiped,
00:43:24.100 --> 00:43:27.570
which is a
parallelepiped-- well,
00:43:27.570 --> 00:43:30.200
in the two-dimensional
case, it's a parallelogram--
00:43:30.200 --> 00:43:37.060
which is spanned by v1 and
v2 or these v's, although you
00:43:37.060 --> 00:43:38.780
have more choices, right?
00:43:38.780 --> 00:43:41.530
So you could have
chosen a different set
00:43:41.530 --> 00:43:43.162
of generating vectors.
00:43:43.162 --> 00:43:45.370
For example, you could have
chosen these two vectors,
00:43:45.370 --> 00:43:48.280
and they also generate
the same lattice.
00:43:48.280 --> 00:43:50.170
And that's also a
fundamental parallelepiped.
00:43:50.170 --> 00:43:51.628
And they will have
the same volume.
00:43:54.120 --> 00:43:56.290
You can make some wrong
choices, and then they
00:43:56.290 --> 00:43:57.760
will not have the right volume.
00:43:57.760 --> 00:44:08.630
So if you had chosen
these two, so this
00:44:08.630 --> 00:44:13.936
is not a fundamental
parallelepiped.
00:44:18.040 --> 00:44:19.160
Great.
00:44:19.160 --> 00:44:20.535
So let me give
you some examples.
00:44:23.610 --> 00:44:31.330
The simplest lattice is just
the integer lattice, Zd,
00:44:31.330 --> 00:44:34.150
which has determinant 1.
00:44:40.090 --> 00:44:41.860
If I'm in the
complex plane, which
00:44:41.860 --> 00:44:50.790
is viewed as two-dimensional
real plane, then if I take,
00:44:50.790 --> 00:44:59.710
let's say, omega being
the 3rd root of unity,
00:44:59.710 --> 00:45:01.370
I have a triangular lattice.
00:45:08.240 --> 00:45:10.960
And the fundamental
parallelepiped
00:45:10.960 --> 00:45:14.290
of this lattice,
that's one example.
00:45:14.290 --> 00:45:16.910
And you can evaluate
its determinant
00:45:16.910 --> 00:45:20.838
as the area of
that parallelogram.
00:45:24.120 --> 00:45:28.540
If I take two nonlinearly
independent vectors--
00:45:28.540 --> 00:45:32.250
so for example, if
I'm in one dimension
00:45:32.250 --> 00:45:35.970
and I look at the integer
span of 1 and root 2,
00:45:35.970 --> 00:45:37.020
this is not a lattice.
00:45:45.732 --> 00:45:49.440
Now, the next definition
will initially
00:45:49.440 --> 00:45:50.610
be slightly confusing.
00:45:50.610 --> 00:45:52.920
But I will explain
it through an example
00:45:52.920 --> 00:45:56.940
or at least try to help you
visualize what's going on.
00:45:56.940 --> 00:46:03.720
So if I give you a centrally
symmetric convex body--
00:46:03.720 --> 00:46:08.360
"centrally symmetric" means
that k equals to minus k.
00:46:08.360 --> 00:46:14.780
So centrally symmetric
convex body, OK.
00:46:14.780 --> 00:46:20.640
So here, centrally
symmetric is x in k
00:46:20.640 --> 00:46:22.510
if and only if minus x is in k.
00:46:25.760 --> 00:46:30.610
And I'm in d dimensions.
00:46:30.610 --> 00:46:40.490
Let me define the i-th
successive minimum
00:46:40.490 --> 00:46:44.500
to be lambda i.
00:46:44.500 --> 00:46:53.180
OK, so i-th successive minimum
of k with respect to lambda
00:46:53.180 --> 00:47:04.450
to be the infimum of all
non-negative lambda such that
00:47:04.450 --> 00:47:14.810
the dimension of the span of the
intersection of lambda k and--
00:47:14.810 --> 00:47:19.190
well, little o lambda
k and the lattice--
00:47:19.190 --> 00:47:24.772
has dimension at least i.
00:47:24.772 --> 00:47:25.700
OK.
00:47:25.700 --> 00:47:26.450
So let me explain.
00:47:33.380 --> 00:47:35.490
I start with a lattice.
00:47:35.490 --> 00:47:38.450
So I start with some lattice.
00:47:38.450 --> 00:47:40.280
And I have some convex body.
00:47:45.110 --> 00:47:46.450
So this is 0, let's say.
00:47:49.110 --> 00:47:55.040
So I have some convex body,
a centrally symmetric convex
00:47:55.040 --> 00:47:56.578
body like that.
00:47:56.578 --> 00:47:58.120
I initially could
be bigger, as well,
00:47:58.120 --> 00:48:02.650
but that's scale it so that
it's quite small initially.
00:48:02.650 --> 00:48:11.740
And let's consider an animation
where I look at lambda k
00:48:11.740 --> 00:48:17.710
where k lambda goes
from 0 to infinity.
00:48:17.710 --> 00:48:18.580
This is k.
00:48:18.580 --> 00:48:22.590
So initially, lambda
k is very, very small.
00:48:22.590 --> 00:48:24.450
And I imagine it growing.
00:48:24.450 --> 00:48:27.040
It gets bigger and
bigger and bigger.
00:48:27.040 --> 00:48:29.500
So it gets bigger and bigger.
00:48:29.500 --> 00:48:32.230
And let's think
about the first time
00:48:32.230 --> 00:48:36.760
that this growing body
hits a lattice point,
00:48:36.760 --> 00:48:39.310
a non-zero lattice point.
00:48:39.310 --> 00:48:42.930
At that point, I
freeze the animation.
00:48:42.930 --> 00:48:46.826
And I record this vector.
00:48:46.826 --> 00:48:58.280
I record this vector where
I've hit a lattice point.
00:48:58.280 --> 00:48:59.790
And now I continue
the animation.
00:48:59.790 --> 00:49:01.498
It's going to keep on
growing and growing
00:49:01.498 --> 00:49:05.110
and growing until when I
hit a vector in a direction
00:49:05.110 --> 00:49:07.560
I haven't seen before.
00:49:07.560 --> 00:49:08.980
So it's going to keep growing.
00:49:08.980 --> 00:49:14.950
And then the next time I hit
a vector in a new direction,
00:49:14.950 --> 00:49:18.450
I stop the animation.
00:49:18.450 --> 00:49:21.540
And I look at the other vector.
00:49:21.540 --> 00:49:27.000
So I keep growing this ball
until I hit new vectors,
00:49:27.000 --> 00:49:29.200
keep growing this convex body.
00:49:29.200 --> 00:49:41.920
So for example, if your initial
convex body is very elongated,
00:49:41.920 --> 00:49:43.050
if that's your k--
00:49:43.050 --> 00:49:44.820
so you keep growing, growing--
00:49:44.820 --> 00:49:50.610
you might initially
hit that vector.
00:49:50.610 --> 00:49:52.280
And then you keep on growing it.
00:49:52.280 --> 00:49:54.020
And the next vector
you hit might still
00:49:54.020 --> 00:49:55.850
be in the same direction.
00:49:55.850 --> 00:49:57.700
But I don't count it.
00:49:57.700 --> 00:49:59.450
I don't stop the
animation here, because I
00:49:59.450 --> 00:50:01.070
didn't see a new direction yet.
00:50:01.070 --> 00:50:05.450
I only stop the animation
when I see a new direction.
00:50:05.450 --> 00:50:10.410
So I keep growing until
I see a new direction.
00:50:10.410 --> 00:50:13.480
And I stop the animation there.
00:50:13.480 --> 00:50:17.410
So think about this growing
body, and stop in every place
00:50:17.410 --> 00:50:22.000
when you see a new direction
contained in your lambda k.
00:50:22.000 --> 00:50:26.000
And the places where
you stop the animations,
00:50:26.000 --> 00:50:31.292
they're the successive
minimum of k.
00:50:31.292 --> 00:50:32.276
Yeah?
00:50:32.276 --> 00:50:36.002
AUDIENCE: Is this defined
if i is greater than d?
00:50:36.002 --> 00:50:38.210
YUFEI ZHAO: Is this defined
when i is greater than d?
00:50:38.210 --> 00:50:38.710
No.
00:50:38.710 --> 00:50:44.880
So you only have exactly
d successive minimum.
00:50:49.040 --> 00:50:52.180
Now, sometimes you might
see two new directions
00:50:52.180 --> 00:50:53.620
at the same time.
00:50:53.620 --> 00:50:55.240
That's OK.
00:50:55.240 --> 00:50:58.393
But once you exhaust
all d directions,
00:50:58.393 --> 00:51:00.560
then there's no more new
directions you can explore.
00:51:03.230 --> 00:51:10.680
We also consider the
vectors that you see.
00:51:10.680 --> 00:51:15.470
So let me also call
these so that we can--
00:51:15.470 --> 00:51:21.570
OK, so we can select
these lattice vectors bi.
00:51:21.570 --> 00:51:26.140
I am going to use
underscore to denote.
00:51:26.140 --> 00:51:29.650
So I'm going to use this
underline to denote boldface.
00:51:29.650 --> 00:51:38.870
So it's a vector bi, which
is in, basically, this.
00:51:38.870 --> 00:51:42.290
You should think of bi as
the new vector that you see.
00:51:42.290 --> 00:51:53.240
And it will have the property
such that b1 through bd
00:51:53.240 --> 00:52:03.210
form a basis of Rd.
00:52:03.210 --> 00:52:05.740
So I keep growing
this convex body.
00:52:05.740 --> 00:52:09.050
When I see a vector in a new
direction, I record lambda.
00:52:09.050 --> 00:52:11.402
And I record the vector bi.
00:52:11.402 --> 00:52:13.510
I keep on going,
keep going, keep
00:52:13.510 --> 00:52:17.380
going until I exhaust
all d directions.
00:52:17.380 --> 00:52:22.195
I call these b's the
directional basis.
00:52:28.742 --> 00:52:29.710
OK.
00:52:29.710 --> 00:52:30.743
Any questions?
00:52:35.473 --> 00:52:37.850
All right.
00:52:37.850 --> 00:52:40.970
So the result from the
geometry of numbers
00:52:40.970 --> 00:52:43.310
that we're going to
need is something called
00:52:43.310 --> 00:52:44.870
Minkowski's second theorem.
00:52:56.150 --> 00:52:59.330
So Minkowski's
second theorem says
00:52:59.330 --> 00:53:05.800
that if you have
lambda, a lattice,
00:53:05.800 --> 00:53:15.580
in Rd and k, a centrally
symmetric body, also in Rd,
00:53:15.580 --> 00:53:19.200
such that lambda
1 through lambda
00:53:19.200 --> 00:53:28.560
d are the successive minima
of k with respect to lambda,
00:53:28.560 --> 00:53:33.580
then one has the inequality
lambda 1, lambda 2.
00:53:33.580 --> 00:53:35.850
So the product of
these successive minima
00:53:35.850 --> 00:53:40.710
times the volume of k
is upper bounded by 2
00:53:40.710 --> 00:53:44.520
to the d times the
determinant of lambda.
00:53:48.000 --> 00:53:54.160
For example, and here is
a very easy case of this
00:53:54.160 --> 00:54:02.960
Minkowski's second theorem, if
your k is an axis-aligned box--
00:54:02.960 --> 00:54:12.240
namely, it is a box where the
width in the i-th direction is
00:54:12.240 --> 00:54:14.220
2 over lambda i--
00:54:22.380 --> 00:54:28.230
so then you see that the
successive minima of this box
00:54:28.230 --> 00:54:31.880
are exactly the lambda i's.
00:54:31.880 --> 00:54:34.760
And you can check that for--
00:54:34.760 --> 00:54:37.288
this inequality is
actually an equality.
00:54:42.410 --> 00:54:42.910
OK.
00:54:42.910 --> 00:54:47.110
So actually, in this
case, lambda, the lattice,
00:54:47.110 --> 00:54:49.150
is the integer lattice.
00:54:49.150 --> 00:54:52.150
Now, this is a pretty easy case
of Minkowski's second theorem.
00:54:52.150 --> 00:54:55.900
But the general case, which
we're not going to prove,
00:54:55.900 --> 00:55:00.030
is actually quite subtle.
00:55:00.030 --> 00:55:02.270
I mean, the proof
itself is not so long.
00:55:02.270 --> 00:55:05.810
It's worth looking up and trying
to see what the proof is about.
00:55:05.810 --> 00:55:08.095
But it's actually
rather counterintuitive
00:55:08.095 --> 00:55:08.720
to think about.
00:55:08.720 --> 00:55:11.743
It's one of those theorems
where you sit down
00:55:11.743 --> 00:55:12.910
for half an hour or an hour.
00:55:12.910 --> 00:55:13.520
You're trying to prove.
00:55:13.520 --> 00:55:15.560
You think you might have
come up with a proof.
00:55:15.560 --> 00:55:18.520
And then on closer
examination, it'll
00:55:18.520 --> 00:55:21.425
be very likely that you
made some very subtle error.
00:55:21.425 --> 00:55:25.550
So it's not so easy to
get all the details right.
00:55:25.550 --> 00:55:28.040
And we're going
to skip the proof.
00:55:28.040 --> 00:55:29.936
But any questions
about the statement?
00:55:34.796 --> 00:55:37.240
OK.
00:55:37.240 --> 00:55:39.220
We're going to use
Minkowski's second theorem
00:55:39.220 --> 00:55:44.860
to show that a large Bohr
set contains a large GAP.
00:55:58.760 --> 00:56:05.530
And specifically, we will
prove that every Bohr
00:56:05.530 --> 00:56:21.910
set of dimension d
and width epsilon--
00:56:21.910 --> 00:56:25.000
epsilon is between 0 and 1--
00:56:25.000 --> 00:56:42.680
in Z mod nZ contains a proper
GAP with dimension at most d
00:56:42.680 --> 00:56:48.650
and size at least
this quantity, which
00:56:48.650 --> 00:56:53.720
is epsilon divided by d raised
to the power of d fraction
00:56:53.720 --> 00:56:54.820
of the cyclic group.
00:57:01.570 --> 00:57:05.100
So just to step
back a bit and see
00:57:05.100 --> 00:57:07.530
where we're going,
from everything
00:57:07.530 --> 00:57:11.670
that we've done earlier, we
conclude that 2A minus 2A
00:57:11.670 --> 00:57:13.770
contains a large Bohr set.
00:57:13.770 --> 00:57:15.360
Here, epsilon is 1/4.
00:57:15.360 --> 00:57:17.160
So epsilon is a constant.
00:57:17.160 --> 00:57:21.620
And R is also going
to be a constant.
00:57:21.620 --> 00:57:23.460
It's depending on the
doubling constant.
00:57:26.050 --> 00:57:27.870
And this proposition
will tell us
00:57:27.870 --> 00:57:31.080
that inside this 2A
minus 2A, we will
00:57:31.080 --> 00:57:33.910
be able to find a very
large, proper GAP.
00:57:33.910 --> 00:57:36.150
So "proper" means that in
this generalized arithmetic
00:57:36.150 --> 00:57:38.860
progression, all the
individual terms are distinct,
00:57:38.860 --> 00:57:40.360
or you don't have collisions.
00:57:40.360 --> 00:57:42.540
So you're going to find
this proper GAP that
00:57:42.540 --> 00:57:47.550
is constant dimension
and at least
00:57:47.550 --> 00:57:52.110
a constant fraction of the
size of the group, so pretty
00:57:52.110 --> 00:57:53.000
large GAP.
00:58:01.490 --> 00:58:05.540
To find this GAP, we
will set up a lattice
00:58:05.540 --> 00:58:07.430
and apply Minkowski's
second theorem.
00:58:13.540 --> 00:58:19.330
Suppose the Bohr
set is given by R
00:58:19.330 --> 00:58:21.450
where the individual
elements, I'm
00:58:21.450 --> 00:58:26.340
going to denote by little
r1 through little rd.
00:58:26.340 --> 00:58:33.960
And let uppercase lambda be
a lattice explicitly given
00:58:33.960 --> 00:58:35.880
as follows.
00:58:35.880 --> 00:58:52.190
It consists of all points in
Rd that are congruent mod 1
00:58:52.190 --> 00:59:05.840
to some integer multiple of the
vector r1 over n, r2 over n,
00:59:05.840 --> 00:59:11.880
through rd over n,
so congruent mod 1.
00:59:14.620 --> 00:59:18.910
So for example, in two
dimensions, which is all
00:59:18.910 --> 00:59:27.950
I can draw on the board,
if r1 and r2 are 1 and 3
00:59:27.950 --> 00:59:33.260
and n equals to
5, then basically,
00:59:33.260 --> 00:59:43.090
what we're going to have is
a refinement of the integer
00:59:43.090 --> 00:59:50.960
lattice, where this box is
going to be the integer lattice.
00:59:50.960 --> 00:59:54.320
And I'm going to tell you some
additional lattice vectors.
00:59:54.320 --> 00:59:59.370
And here, it's going to repeat,
or it's going to tile all over.
00:59:59.370 --> 01:00:04.390
So I start with 1, 3.
01:00:04.390 --> 01:00:06.520
And I look at multiples of it.
01:00:06.520 --> 01:00:08.930
But I mod 1.
01:00:08.930 --> 01:00:15.650
So I would end up with these
points and then repeat it.
01:00:26.162 --> 01:00:27.120
And so you would have--
01:00:32.180 --> 01:00:33.420
so that's the lattice.
01:00:33.420 --> 01:00:37.810
So you have this
lattice, lambda.
01:00:37.810 --> 01:00:39.460
What is the volume?
01:00:39.460 --> 01:00:43.120
What is the determinant
of this lattice?
01:00:43.120 --> 01:00:44.750
So the determinant
of the lattice,
01:00:44.750 --> 01:00:50.900
remember, is the volume of its
fundamental parallelepiped.
01:00:50.900 --> 01:00:58.480
So I claim that the determinant
is exactly 1 over n.
01:00:58.480 --> 01:01:00.420
There are a few
ways to see this.
01:01:00.420 --> 01:01:04.300
So one is that, originally,
I had the integer
01:01:04.300 --> 01:01:06.170
lattice as determinant 1.
01:01:06.170 --> 01:01:07.870
And now I put--
01:01:07.870 --> 01:01:09.940
instead of one point,
I have endpoints
01:01:09.940 --> 01:01:12.070
in each original parallelepiped.
01:01:12.070 --> 01:01:16.030
So the determinant has to
go down by a factor of n.
01:01:16.030 --> 01:01:20.890
Or you can construct an explicit
fundamental parallelepiped
01:01:20.890 --> 01:01:21.790
like that.
01:01:21.790 --> 01:01:25.186
And then you use
base times height.
01:01:25.186 --> 01:01:25.686
OK.
01:01:33.110 --> 01:01:36.750
We're going to apply
Minkowski's second theorem.
01:01:36.750 --> 01:01:38.450
And I will need to tell you--
01:01:38.450 --> 01:01:40.950
I don't need the definition
of Bohr set up there.
01:01:40.950 --> 01:01:45.660
So I want to tell you what
to use as the convex body.
01:01:57.760 --> 01:02:00.580
The convex body that
we're going to use
01:02:00.580 --> 01:02:11.280
is k being this box
of width 2 epsilon.
01:02:11.280 --> 01:02:12.780
So that's the lattice.
01:02:12.780 --> 01:02:14.990
That's the convex body.
01:02:14.990 --> 01:02:17.910
And we're going to apply
Minkowski's second theorem.
01:02:17.910 --> 01:02:25.370
So let's let little lowercase
lambda 1 through lambda d--
01:02:25.370 --> 01:02:28.470
so n is d--
01:02:28.470 --> 01:02:39.290
be the successive minima
of k with respect to lambda
01:02:39.290 --> 01:02:47.800
and b be the
directional vectors,
01:02:47.800 --> 01:02:52.750
the rational basis corresponding
to those successive minima.
01:02:56.640 --> 01:03:03.830
I claim that the
L-infinity norm of bj
01:03:03.830 --> 01:03:12.310
is at most lambda j
epsilon for each j.
01:03:12.310 --> 01:03:16.080
And this is basically
because of the definition.
01:03:16.080 --> 01:03:18.615
I mean, if you look at the
definition of successive minima
01:03:18.615 --> 01:03:28.900
and directional
basis, this is k.
01:03:28.900 --> 01:03:34.020
I grow k, grow it by
a factor of lambda j.
01:03:34.020 --> 01:03:39.030
And that's the first
point when I see b sub j.
01:03:39.030 --> 01:03:42.030
So every coordinate
of b sub j has
01:03:42.030 --> 01:03:45.230
to be at most this
quantity in absolute value.
01:03:50.470 --> 01:03:55.900
So now let me denote
uppercase L sub j
01:03:55.900 --> 01:04:02.380
to be 1 over lambda
jd rounded up.
01:04:07.510 --> 01:04:16.200
And I claim that if little
l is less than big L--
01:04:16.200 --> 01:04:26.710
lj, Lj-- then little lj--
01:04:26.710 --> 01:04:33.070
OK, so if I dilate the bj
vector by factor little l,
01:04:33.070 --> 01:04:38.980
so if I plug it in and just
look at these two inequalities,
01:04:38.980 --> 01:04:42.280
I obtain an upper
bound of epsilon
01:04:42.280 --> 01:04:50.345
over d on the L-infinity
norm of lj bj,
01:04:50.345 --> 01:04:55.100
so just looking at this bound
here and the size of lj.
01:04:58.650 --> 01:05:14.020
And if this holds for
all j, then summing up
01:05:14.020 --> 01:05:16.240
all of these individual
inequalities,
01:05:16.240 --> 01:05:28.770
we find that the sum of
these lj bj's is at most
01:05:28.770 --> 01:05:30.700
epsilon in L-infinity norm.
01:05:47.040 --> 01:05:49.380
So the point here
is that we want
01:05:49.380 --> 01:05:52.980
to find the GAP
in this Bohr set.
01:05:52.980 --> 01:05:56.550
And how does one
think of a Bohr set?
01:05:56.550 --> 01:05:59.730
So it's kind of hard to
imagine, because the Bohr set
01:05:59.730 --> 01:06:01.890
is a subset of Z mod n.
01:06:01.890 --> 01:06:04.140
But the right way to
think about a Bohr set
01:06:04.140 --> 01:06:09.810
is in a higher dimensional lift,
because a Bohr set is defined
01:06:09.810 --> 01:06:18.220
by looking at these numbers
for R different values,
01:06:18.220 --> 01:06:19.510
R different coordinates.
01:06:19.510 --> 01:06:24.540
So we think of each r
as its own coordinate.
01:06:24.540 --> 01:06:28.630
So we think of there
being capital uppercase
01:06:28.630 --> 01:06:31.690
R many coordinates.
01:06:31.690 --> 01:06:36.460
And we want to
consider the set of x's
01:06:36.460 --> 01:06:41.510
so that all the coordinate
values are small.
01:06:41.510 --> 01:06:43.800
So instead of considering
a one-dimensional picture,
01:06:43.800 --> 01:06:46.008
as we do in the Bohr sets,
we're considering a higher
01:06:46.008 --> 01:06:48.030
dimensional or
d-dimensional picture
01:06:48.030 --> 01:06:50.760
and then eventually
projecting what happens up
01:06:50.760 --> 01:06:53.125
there down to this Bohr set.
01:06:53.125 --> 01:06:54.750
So what does Minkowski's
second theorem
01:06:54.750 --> 01:06:56.310
have to do with anything?
01:06:56.310 --> 01:07:02.200
Well, once you have this higher
dimensional lattice, what we're
01:07:02.200 --> 01:07:09.120
going to do is find a large
lattice parallelepiped,
01:07:09.120 --> 01:07:12.360
so a large structure inside
this higher dimensional
01:07:12.360 --> 01:07:15.150
lattice, and then
project it down
01:07:15.150 --> 01:07:19.370
onto one-dimensional Z mod n.
01:07:19.370 --> 01:07:21.340
So this is the process of--
01:07:21.340 --> 01:07:26.000
so you already see some
aspects of a GAP in here.
01:07:26.000 --> 01:07:28.730
So these guys,
they're essentially
01:07:28.730 --> 01:07:33.960
the GAP that we're going
to eventually wish to find.
01:07:33.960 --> 01:07:37.460
And right now, they live in
this higher dimensional lattice.
01:07:37.460 --> 01:07:39.870
But we're going to pull
them down to Z mod n.
01:07:44.480 --> 01:07:46.280
All right.
01:07:46.280 --> 01:07:50.530
Now, where do these
b's come from?
01:07:50.530 --> 01:08:08.950
So each b sub j is congruent to
some x sub j times this vector
01:08:08.950 --> 01:08:20.200
mod 1 where x sub
j is an integer
01:08:20.200 --> 01:08:27.370
between 0 and uppercase N.
01:08:27.370 --> 01:08:34.960
So this inequality
up here, star.
01:08:34.960 --> 01:08:41.859
So the i-th
coordinate for star--
01:08:41.859 --> 01:08:44.189
"coordinate" meaning this
is an L-infinity bound,
01:08:44.189 --> 01:08:47.229
so the i-th coordinate is
upper bounded by epsilon.
01:08:47.229 --> 01:08:50.340
But the i-th coordinate
bound implies
01:08:50.340 --> 01:09:04.840
that if you look at this sum
over here times Ri divided
01:09:04.840 --> 01:09:12.220
by N, this quantity,
whatever it is,
01:09:12.220 --> 01:09:20.810
is very close to an
integer for each i.
01:09:20.810 --> 01:09:24.100
So the i-th coordinates
implies this inequality,
01:09:24.100 --> 01:09:25.354
and it's true for every i.
01:09:28.600 --> 01:09:36.510
Thus what we find is
that the GAP, which you
01:09:36.510 --> 01:09:38.600
already see in this
formula over here--
01:09:38.600 --> 01:09:45.322
so the GAP is given like that.
01:09:50.640 --> 01:09:54.658
So this GAP is contained
in the Bohr set.
01:10:04.130 --> 01:10:10.390
So we found a large
structure in the lattice.
01:10:10.390 --> 01:10:13.340
But the lattice came
from this construction,
01:10:13.340 --> 01:10:16.240
which was directly
motivated by the Bohr set.
01:10:16.240 --> 01:10:18.990
So we find a large
GAP in the Bohr set.
01:10:18.990 --> 01:10:24.100
Well, we haven't shown yet it
is large or that it is proper.
01:10:24.100 --> 01:10:25.920
So we need to check
those two things.
01:10:40.960 --> 01:10:45.460
To check that this GAP
that we found is large,
01:10:45.460 --> 01:10:47.860
we're going to apply
Minkowski's second theorem.
01:10:53.950 --> 01:10:58.215
Let's check GAP is large.
01:11:01.440 --> 01:11:11.000
So by Minkowski's
second theorem,
01:11:11.000 --> 01:11:15.710
we find that the size of the
GAP, which is, by definition,
01:11:15.710 --> 01:11:19.150
the product of these
upper case L's--
01:11:19.150 --> 01:11:22.090
so if you look at how the
uppercase L's is defined,
01:11:22.090 --> 01:11:24.640
you see that this
quantity is at least 1
01:11:24.640 --> 01:11:30.230
over the product of the
successive minima times
01:11:30.230 --> 01:11:33.030
denominator d to the d.
01:11:33.030 --> 01:11:36.530
And now we apply
Minkowski's second.
01:11:36.530 --> 01:11:42.250
And we find that this
quantity is at least
01:11:42.250 --> 01:11:47.800
the volume of k
divided by 2 to the d
01:11:47.800 --> 01:11:55.010
times the determinant of the
lattice times d to the d.
01:11:55.010 --> 01:11:59.450
But we saw what is the
determinant of the lattice.
01:11:59.450 --> 01:12:06.210
It is 1 over N. You have
d to the d, 2 to the d.
01:12:06.210 --> 01:12:09.320
And the volume of k,
well, k is just that box.
01:12:09.320 --> 01:12:13.830
So the volume of k is
2 epsilon raised to d.
01:12:13.830 --> 01:12:16.470
So putting everything
together, we
01:12:16.470 --> 01:12:23.940
find that the size of this
GAP is the claimed quantity.
01:12:23.940 --> 01:12:26.352
It's a constant fraction
of the entire group.
01:12:29.672 --> 01:12:31.880
The second thing that we
need to check is properness.
01:12:38.090 --> 01:12:40.510
So what does it
mean to be proper?
01:12:40.510 --> 01:12:42.940
So we just want to
know that you don't
01:12:42.940 --> 01:12:45.250
have two different
ways of representing
01:12:45.250 --> 01:12:48.490
the same term in the GAP.
01:12:48.490 --> 01:12:58.480
So if I have the
following congruence,
01:12:58.480 --> 01:13:07.180
so if this combination
of the x's is
01:13:07.180 --> 01:13:21.700
congruent to a different
combination of the x's where
01:13:21.700 --> 01:13:28.240
these little l's
are between 1 and--
01:13:28.240 --> 01:13:31.190
OK, so I want to show-- so
to check that it's proper--
01:13:31.190 --> 01:13:32.780
so we're in Z mod n--
01:13:32.780 --> 01:13:35.360
we just need to check
that if this holds,
01:13:35.360 --> 01:13:38.690
then all the corresponding
little l's must
01:13:38.690 --> 01:13:42.530
be the same as their primes.
01:13:42.530 --> 01:13:49.240
Well, if it is
true, then setting--
01:13:49.240 --> 01:13:50.830
let's go back to the lattice--
01:13:50.830 --> 01:13:59.720
setting the vector b to
be a vector originally
01:13:59.720 --> 01:14:02.400
that corresponds to the
difference of these two
01:14:02.400 --> 01:14:02.900
numbers--
01:14:14.800 --> 01:14:17.580
so if we set b to be the
difference of these two
01:14:17.580 --> 01:14:27.250
numbers, we find that, first
of all, it lies in Z to the d,
01:14:27.250 --> 01:14:32.220
because these two numbers are
congruent to each other mod n.
01:14:41.220 --> 01:14:56.140
And furthermore, the L-infinity
norm of b is upper bounded by--
01:14:58.780 --> 01:15:03.550
I mean, each one of them
has small l-infinity norm.
01:15:03.550 --> 01:15:07.200
And this is some
number that is bounded.
01:15:07.200 --> 01:15:11.000
It's less than uppercase L.
01:15:11.000 --> 01:15:15.730
So the whole thing, this whole
sum, the L-infinity norm,
01:15:15.730 --> 01:15:21.460
cannot be larger than
this quantity over here,
01:15:21.460 --> 01:15:26.190
where I essentially use the
triangle inequality to analyze
01:15:26.190 --> 01:15:27.500
this b term by term.
01:15:30.680 --> 01:15:33.380
All of these numbers
are very small,
01:15:33.380 --> 01:15:42.520
because if you look at what we
saw up there, so the size of b,
01:15:42.520 --> 01:15:47.230
we see that this whole
thing is at most epsilon.
01:15:50.020 --> 01:15:53.370
And epsilon is
strictly less than 1.
01:15:53.370 --> 01:16:01.110
So you have some vector b,
which is an integer vector,
01:16:01.110 --> 01:16:10.400
such that all of its coordinates
have L-infinity norms strictly
01:16:10.400 --> 01:16:12.630
less than 1.
01:16:12.630 --> 01:16:14.920
So that means that
b is equal to 0.
01:16:22.280 --> 01:16:23.590
So b is the 0 vector.
01:16:27.292 --> 01:16:28.890
So b is a zero vector.
01:16:34.180 --> 01:16:39.920
Thus this thing
here equals to 0.
01:16:39.920 --> 01:16:42.760
So this sum here equals to 0.
01:16:42.760 --> 01:16:50.650
And since the bi's
form a basis, we
01:16:50.650 --> 01:16:55.480
find that the li's
and l prime i's
01:16:55.480 --> 01:16:58.371
are equal to each
other for all i.
01:17:01.670 --> 01:17:04.020
And this checks the
properness of this GAP.
01:17:08.520 --> 01:17:09.020
Yeah.
01:17:09.020 --> 01:17:10.740
So this argument, it's not hard.
01:17:10.740 --> 01:17:13.350
But you need to
check the details.
01:17:13.350 --> 01:17:17.640
So you need to wrap your
mind around changing
01:17:17.640 --> 01:17:20.280
from working in a higher
dimensional lattice setting
01:17:20.280 --> 01:17:23.520
to going back down to Z mod n.
01:17:23.520 --> 01:17:26.340
And the main takeaway
here is that the right way
01:17:26.340 --> 01:17:29.780
to think about a Bohr set
is to not stay in Z mod n
01:17:29.780 --> 01:17:32.970
but to think about what
happens in d-dimensional space
01:17:32.970 --> 01:17:35.070
where d is the dimension
of the Bohr set.
01:17:39.520 --> 01:17:40.020
OK.
01:17:40.020 --> 01:17:42.240
So now we have pretty
much all the ingredients
01:17:42.240 --> 01:17:45.240
that we need to prove
Freiman's theorem.
01:17:45.240 --> 01:17:48.190
And that's what we'll do at
the beginning of next lecture.
01:17:48.190 --> 01:17:50.550
We'll conclude the proof
of Freiman's theorem.
01:17:50.550 --> 01:17:53.505
And then I'll tell you also
about an important conjecture
01:17:53.505 --> 01:17:56.490
in additive combinatorics called
a polynomial Freiman-Ruzsa
01:17:56.490 --> 01:17:59.100
conjecture, which
many people think
01:17:59.100 --> 01:18:02.730
is the most important
open conjecture
01:18:02.730 --> 01:18:05.660
in additive combinatorics.