WEBVTT
00:00:18.502 --> 00:00:20.460
YUFEI ZHAO: The goal for
the next few lecturers
00:00:20.460 --> 00:00:23.905
is to prove Freiman's theorem,
which we discussed last time.
00:00:23.905 --> 00:00:27.410
And so we started with
this tool that we proved
00:00:27.410 --> 00:00:29.820
called the Plunnecke-Ruzsa
inequality,
00:00:29.820 --> 00:00:33.540
which tells us that if you have
a set A now in an arbitrary
00:00:33.540 --> 00:00:37.890
abelian group, and if A
has controlled doubling,
00:00:37.890 --> 00:00:40.530
it has bounded
doubling, then all
00:00:40.530 --> 00:00:47.160
the further iterated sumsets
have bounded growth, as well.
00:00:47.160 --> 00:00:49.130
So this is what we
proved last time.
00:00:49.130 --> 00:00:51.900
And so we're going to be using
the Plunnecke-Ruzsa inequality
00:00:51.900 --> 00:00:53.970
many times.
00:00:53.970 --> 00:00:56.910
But there are some other tools
I need to tell you about.
00:01:00.250 --> 00:01:03.830
So the next tool is known
as Ruzsa covering lemma.
00:01:12.090 --> 00:01:12.660
All right.
00:01:12.660 --> 00:01:14.285
So let me first give
you the statement,
00:01:14.285 --> 00:01:17.222
and then I explain
some intuition.
00:01:17.222 --> 00:01:19.680
I think the technique is more
important than the statement.
00:01:19.680 --> 00:01:22.560
But in any case,
here's what it says.
00:01:22.560 --> 00:01:28.010
If you have X and B--
00:01:28.010 --> 00:01:30.480
they're subsets of some
arbitrary abelian group--
00:01:38.370 --> 00:01:46.500
if you have the inequality that
X plus B is at most K times
00:01:46.500 --> 00:01:49.320
the size of B-- so the size
of X plus B is at most K times
00:01:49.320 --> 00:01:50.610
the size of B--
00:01:50.610 --> 00:02:03.870
then there exists
some subset T of X,
00:02:03.870 --> 00:02:08.430
with the size of T
being, at most, K,
00:02:08.430 --> 00:02:20.250
such that X is contained
in T plus B minus B. So
00:02:20.250 --> 00:02:22.200
that's the statement of
Ruzsa covering lemma.
00:02:22.200 --> 00:02:25.710
So let me play to
explain what it is about.
00:02:25.710 --> 00:02:32.250
So the idea here is that, if
you're in a situation where
00:02:32.250 --> 00:02:34.860
if it looks like--
00:02:37.780 --> 00:02:40.770
so you should think
of B as a ball.
00:02:40.770 --> 00:02:44.180
So if it looks
like X plus B might
00:02:44.180 --> 00:02:58.260
be coverable by K
translates of B--
00:02:58.260 --> 00:03:04.080
so here, you're supposed
to think of B like a ball
00:03:04.080 --> 00:03:05.520
in the metric space--
00:03:05.520 --> 00:03:13.020
then X is actually coverable.
00:03:16.490 --> 00:03:19.120
So if it looks like,
meaning just by size
00:03:19.120 --> 00:03:24.320
alone, by size info alone--
00:03:24.320 --> 00:03:26.800
so just based on the
size, if it looks
00:03:26.800 --> 00:03:31.390
like X plus B might be coverable
by K different translates of B,
00:03:31.390 --> 00:03:34.750
then actually X is
coverable, but you
00:03:34.750 --> 00:03:45.960
have to use slightly larger
balls by K copies of B minus B.
00:03:45.960 --> 00:03:49.050
So you should think of B minus
B as slightly larger balls
00:03:49.050 --> 00:03:50.090
than B itself.
00:03:50.090 --> 00:03:52.710
So if B were an actual
ball in Euclidean space,
00:03:52.710 --> 00:03:55.200
then B minus B is the same
ball with twice the radius.
00:04:03.160 --> 00:04:07.530
And so Ruzsa covering lemma
is a really important tool.
00:04:07.530 --> 00:04:09.000
The proof is not very long.
00:04:09.000 --> 00:04:12.010
And it's important to understand
the idea of this proof.
00:04:12.010 --> 00:04:14.820
So this is a proof not just
in additive combinatorics
00:04:14.820 --> 00:04:17.160
but something that happens--
00:04:17.160 --> 00:04:19.550
it's a standard
idea in analysis.
00:04:19.550 --> 00:04:22.560
So it's very important
to understand this idea.
00:04:22.560 --> 00:04:24.350
And the key idea--
00:04:24.350 --> 00:04:26.850
here, I think the proof is more
important than the statement
00:04:26.850 --> 00:04:28.110
up there--
00:04:28.110 --> 00:04:36.840
the key idea here is that if
you want to produce a covering,
00:04:36.840 --> 00:04:40.635
one way to produce a covering
is to take a maximal packing.
00:04:45.960 --> 00:04:55.640
A maximal packing
using with balls,
00:04:55.640 --> 00:05:07.260
for instance, implies a covering
with balls twice as large.
00:05:11.940 --> 00:05:14.910
So let me illustrate
that with a picture.
00:05:14.910 --> 00:05:20.920
Suppose you have some
space that I want to cover.
00:05:20.920 --> 00:05:24.290
But you get to use,
let's say, unit balls.
00:05:24.290 --> 00:05:27.590
So how can I make
sure I can cover
00:05:27.590 --> 00:05:29.000
the space using unit balls?
00:05:29.000 --> 00:05:32.060
And I don't want to use
too many unit balls.
00:05:32.060 --> 00:05:45.260
So what you can do is, let me
start by a maximal set of unit
00:05:45.260 --> 00:05:47.250
balls, so with centers--
00:05:47.250 --> 00:05:53.060
so it's now half unit
balls, so the radius is 1/2.
00:05:58.450 --> 00:06:03.170
So I put in as many as I can so
that I cannot put in any more.
00:06:03.170 --> 00:06:05.080
So that's what maximal means--
00:06:05.080 --> 00:06:07.360
"mal" here doesn't mean
the maximum number.
00:06:07.360 --> 00:06:11.350
Although, if you put in the
maximum number, that's also OK.
00:06:11.350 --> 00:06:14.560
But maximal means that I just
cannot put in any more balls
00:06:14.560 --> 00:06:16.810
that are not overlapping.
00:06:16.810 --> 00:06:19.600
If I have this
configuration, now what I do
00:06:19.600 --> 00:06:22.390
is I double the radii
of all the balls.
00:06:28.710 --> 00:06:31.178
So whoever takes today's notes
will have a fun time drawing
00:06:31.178 --> 00:06:31.720
this picture.
00:06:36.310 --> 00:06:38.935
So this has to be a covering
of the original space.
00:06:41.670 --> 00:06:44.520
Because if you had
missed some point,
00:06:44.520 --> 00:06:47.500
I could have put a blue ball in.
00:06:47.500 --> 00:06:48.161
Yes?
00:06:48.161 --> 00:06:51.065
AUDIENCE: What if a space
has some narrow portion?
00:06:51.065 --> 00:06:53.690
YUFEI ZHAO: Question-- what if
a space has some narrow portion?
00:06:53.690 --> 00:06:54.990
It doesn't matter.
00:06:54.990 --> 00:06:58.310
If you formulate
this correctly--
00:06:58.310 --> 00:07:02.770
if you miss some
point-- so here,
00:07:02.770 --> 00:07:06.635
it depends on how you
formulate this covering.
00:07:06.635 --> 00:07:15.130
The point is that, if you
take a maximal set of points,
00:07:15.130 --> 00:07:18.730
when you expand, you have
to cover the whole space.
00:07:18.730 --> 00:07:20.920
Because if you
missed some point--
00:07:20.920 --> 00:07:23.655
so imagine if you had--
00:07:23.655 --> 00:07:25.030
for example,
suppose you missed--
00:07:28.700 --> 00:07:31.490
suppose you had missed
some point over here.
00:07:31.490 --> 00:07:34.510
Then I could have
put an extra ball in.
00:07:34.510 --> 00:07:38.070
So if you had missed
some point over here,
00:07:38.070 --> 00:07:42.730
that means that I should have
been able to put in that ball
00:07:42.730 --> 00:07:45.030
there initially.
00:07:45.030 --> 00:07:48.930
So this is a very simple idea,
but a very powerful idea.
00:07:48.930 --> 00:07:52.200
And it comes up all the time
in analysis and geometry.
00:07:52.200 --> 00:07:54.930
And it also comes up here.
00:07:54.930 --> 00:07:56.640
So let's do the actual proof.
00:08:01.210 --> 00:08:19.160
So let me let T be a subset
of X be a maximal subset of X,
00:08:19.160 --> 00:08:25.520
such that the sets
little t plus B are
00:08:25.520 --> 00:08:35.659
disjoint for all elements
little t of a set big T.
00:08:35.659 --> 00:08:37.299
So it's like this picture.
00:08:37.299 --> 00:08:41.350
I pick a subset of X so
that if I center balls
00:08:41.350 --> 00:08:45.700
around these t's, then
these translates of B's are
00:08:45.700 --> 00:08:46.700
destroyed.
00:08:46.700 --> 00:08:50.990
Then you put in a
maximal such set.
00:08:50.990 --> 00:08:58.040
So due to the
disjointness, we find
00:08:58.040 --> 00:09:04.990
that the product of
the sizes of t and B
00:09:04.990 --> 00:09:10.985
equals to the size of the sum,
because there's no overlaps.
00:09:13.780 --> 00:09:19.130
But t plus B has size
at most X plus B,
00:09:19.130 --> 00:09:23.750
because T is a subset of X.
But also, from the assumption
00:09:23.750 --> 00:09:28.310
we knew that X is--
00:09:28.310 --> 00:09:35.870
size of x plus B is upper
bounded by the size B times K.
00:09:35.870 --> 00:09:40.790
So in particular, we
get that the size of T
00:09:40.790 --> 00:09:47.480
is at most K. So in
other words, over here,
00:09:47.480 --> 00:09:50.810
the number of blue balls you can
control by simply the volume.
00:09:53.890 --> 00:10:13.350
Now, since T is maximal, we
have that for every little x,
00:10:13.350 --> 00:10:24.960
there exists some little t
such that the translate of B
00:10:24.960 --> 00:10:28.680
given by x intersects one
of my children translates.
00:10:31.520 --> 00:10:34.360
Because if this were
not true, then I
00:10:34.360 --> 00:10:41.000
could have put in an
extra translate of B.
00:10:41.000 --> 00:10:48.800
So in other words, there exist
two elements, b and b prime,
00:10:48.800 --> 00:10:54.110
such that t plus b
equals to x plus b prime.
00:10:54.110 --> 00:11:09.040
And hence, x lies in little
t plus B minus B, which
00:11:09.040 --> 00:11:20.600
implies that the set X lies
in T plus B minus B. OK.
00:11:20.600 --> 00:11:23.120
So that's the Ruzsa
covering lemma.
00:11:23.120 --> 00:11:25.520
So it's an execution
of this idea
00:11:25.520 --> 00:11:30.080
I mentioned earlier that
a maximal packing implies
00:11:30.080 --> 00:11:30.830
a good covering.
00:11:33.460 --> 00:11:34.464
Any questions?
00:11:36.960 --> 00:11:37.460
Right.
00:11:37.460 --> 00:11:41.390
So now we have this tool, we
can prove an easier version
00:11:41.390 --> 00:11:45.230
of Freiman's theorem where,
instead of working in integers,
00:11:45.230 --> 00:11:47.430
we're going to work in
the finite field model.
00:11:47.430 --> 00:11:51.088
So usually, it's good to start
with finite field models.
00:11:51.088 --> 00:11:52.130
Things are a bit cleaner.
00:11:52.130 --> 00:11:54.500
And in this case, it
actually only requires
00:11:54.500 --> 00:11:57.290
a subset of the tools that
we need for the full theorem.
00:12:00.880 --> 00:12:02.975
So instead of working
in finite field model,
00:12:02.975 --> 00:12:05.350
we're going to be working in
something just slightly more
00:12:05.350 --> 00:12:05.850
general.
00:12:05.850 --> 00:12:06.850
But it's the same proof.
00:12:10.270 --> 00:12:16.580
So we're going to be working
in a group of bounded exponent.
00:12:16.580 --> 00:12:26.420
Freiman's theorem in
groups of bounded exponent.
00:12:26.420 --> 00:12:32.850
And the word "exponent" in group
theory means the following--
00:12:32.850 --> 00:12:41.440
so the exponent of
an abelian group
00:12:41.440 --> 00:12:52.820
is the smallest positive
integer, if it exists.
00:12:57.490 --> 00:13:04.420
So the smallest positive
integer r so that rx
00:13:04.420 --> 00:13:08.430
equals to 0 for
all x in the group.
00:13:18.300 --> 00:13:22.580
So for example, if you
are working Fp to the n,
00:13:22.580 --> 00:13:24.680
then the exponent is p.
00:13:24.680 --> 00:13:28.560
So every element has--
00:13:28.560 --> 00:13:34.710
if you add to itself r times,
then it vanishes the element.
00:13:37.920 --> 00:13:39.948
The word "exponent" comes from--
00:13:39.948 --> 00:13:41.490
I mean, you can
define the same thing
00:13:41.490 --> 00:13:43.032
for non-abelian
groups, in which case
00:13:43.032 --> 00:13:46.830
you should write this
expression using the exponent.
00:13:46.830 --> 00:13:49.892
Instead of addition,
you have multiplication.
00:13:49.892 --> 00:13:51.350
So that's why it's
called exponent.
00:13:51.350 --> 00:13:54.680
So the name is stuck, even
though we're still working
00:13:54.680 --> 00:13:55.680
on the additive setting.
00:13:58.470 --> 00:14:03.450
We're going to write
this, so the angle
00:14:03.450 --> 00:14:09.340
brackets, to mean the
subgroup generated
00:14:09.340 --> 00:14:15.010
by the subset A, where A is a
subset of the group elements.
00:14:15.010 --> 00:14:20.140
And so the exponent
of a group, then,
00:14:20.140 --> 00:14:22.870
is equal to the
maximum number of--
00:14:26.170 --> 00:14:28.330
so if you pick a
group element, look
00:14:28.330 --> 00:14:31.630
at how many group
elements does it generate.
00:14:31.630 --> 00:14:32.980
So it's equal to the max.
00:14:38.280 --> 00:14:39.148
All right.
00:14:39.148 --> 00:14:40.940
I mean, the example
you can have in mind is
00:14:40.940 --> 00:14:44.390
F2 to the n, which
has exponent 2.
00:14:44.390 --> 00:14:46.830
So in general, we're
going to be looking
00:14:46.830 --> 00:14:49.740
at a special case in a
group with bounded exponent.
00:14:52.350 --> 00:14:54.480
And Freiman's
theorem, in this case,
00:14:54.480 --> 00:15:01.260
is due to Ruzsa, who
showed that if you
00:15:01.260 --> 00:15:21.470
have a finite subset in an
abelian group with exponent r--
00:15:21.470 --> 00:15:22.730
so a finite exponent--
00:15:25.890 --> 00:15:33.422
if A has doubling
constant at most K, then
00:15:33.422 --> 00:15:34.380
what do we want to say?
00:15:34.380 --> 00:15:36.338
We want to say, just like
in Freiman's theorem,
00:15:36.338 --> 00:15:38.370
that if A has bounded
doubling, then
00:15:38.370 --> 00:15:42.770
it is a large portion
of some structured set.
00:15:42.770 --> 00:15:46.922
And here "structured
set" means subgroup.
00:15:46.922 --> 00:15:49.130
Well, if you're going to be
looking at subgroups that
00:15:49.130 --> 00:15:52.610
contain A, you might as well
look at the subgroup generated
00:15:52.610 --> 00:15:58.220
by A. So the claim, then, is
that the subgroup generated
00:15:58.220 --> 00:16:04.280
by A is only a constant
factor bigger than A itself.
00:16:12.610 --> 00:16:14.410
So let me say it again.
00:16:14.410 --> 00:16:17.890
If you have a set A in a
group of bounded exponent,
00:16:17.890 --> 00:16:21.970
and A has bounded
doubling, then conclusion
00:16:21.970 --> 00:16:27.080
is that A is a large
proportion of some subgroup.
00:16:27.080 --> 00:16:28.910
Conversely-- we saw last time--
00:16:28.910 --> 00:16:31.400
if you take a subgroup, it
has doubling constant 1,
00:16:31.400 --> 00:16:34.100
so if you take a positive
proportional constant
00:16:34.100 --> 00:16:37.750
proportion of the subgroup,
it also has bounded doubling.
00:16:37.750 --> 00:16:40.480
And this statement
is in some sense
00:16:40.480 --> 00:16:44.570
the converse of
that observation.
00:16:44.570 --> 00:16:47.240
This bound here is not optimal.
00:16:47.240 --> 00:16:48.920
I'll make some
comments about what
00:16:48.920 --> 00:16:50.073
we know about these bounds.
00:16:50.073 --> 00:16:52.115
But for now, just view
this number as a constant.
00:16:54.750 --> 00:16:56.540
Any questions about
the statement?
00:16:59.830 --> 00:17:01.240
All right.
00:17:01.240 --> 00:17:07.349
So let's prove this
Ruzsa's theorem, giving you
00:17:07.349 --> 00:17:10.740
Freiman's theorem in groups
with bounded exponent.
00:17:10.740 --> 00:17:13.470
So we're going to be applying
the tools we've seen so far,
00:17:13.470 --> 00:17:15.530
starting with Plunnecke-Ruzsa.
00:17:21.650 --> 00:17:25.770
So by Plunnecke-Ruzsa
inequality, we find that--
00:17:25.770 --> 00:17:27.770
so then I'm going to write
down some expression.
00:17:27.770 --> 00:17:29.270
You may wonder, why
this expression?
00:17:29.270 --> 00:17:33.190
Because we're going to apply
covering lemma in a second.
00:17:33.190 --> 00:17:39.930
So this set by Plunnecke-Ruzsa,
its size is bounded,
00:17:39.930 --> 00:17:43.560
because the set can also
be written as 3A minus A,
00:17:43.560 --> 00:17:49.620
so its size is bounded by K to
the fourth times the size of A.
00:17:49.620 --> 00:17:53.112
And so Plunnecke-Ruzsa is
a very nice tool to have.
00:17:53.112 --> 00:17:55.320
It basically tells you, if
you have bounded doubling,
00:17:55.320 --> 00:17:57.630
then all the other iterated
sums are essentially
00:17:57.630 --> 00:17:58.848
controlled in size.
00:17:58.848 --> 00:18:00.140
And then we're using that here.
00:18:03.280 --> 00:18:03.780
OK.
00:18:03.780 --> 00:18:05.447
So now we're in the
setting where we can
00:18:05.447 --> 00:18:08.520
apply the Ruzsa covering lemma.
00:18:08.520 --> 00:18:14.642
So by covering
lemma, we're going
00:18:14.642 --> 00:18:15.850
to apply the covering lemma--
00:18:15.850 --> 00:18:17.990
so using the notation earlier--
00:18:17.990 --> 00:18:29.430
with X equal to 2A minus A and
B equal to A. By covering lemma,
00:18:29.430 --> 00:18:39.858
there exists some T being a
subset of 2A minus A, with T
00:18:39.858 --> 00:18:42.450
not too large--
00:18:42.450 --> 00:18:46.580
because of our
earlier estimate--
00:18:46.580 --> 00:18:58.910
and such that 2A minus A is
contained in T plus A minus A.
00:18:58.910 --> 00:19:01.650
So it's easy to get
lost in the details.
00:19:01.650 --> 00:19:04.830
But what's happening here
is that I start with A,
00:19:04.830 --> 00:19:10.620
and I'm looking at how
big these iterated growing
00:19:10.620 --> 00:19:12.880
sumsets can be.
00:19:12.880 --> 00:19:15.780
And if I keep on
doing this operation,
00:19:15.780 --> 00:19:18.150
like if I just apply
Plunnecke-Ruzsa,
00:19:18.150 --> 00:19:22.380
I can't really control
the iterated sums
00:19:22.380 --> 00:19:23.880
by an absolute bound.
00:19:23.880 --> 00:19:28.930
This bound will keep growing
if I take bigger iterations.
00:19:28.930 --> 00:19:30.660
But Ruzsa covering
lemma gives me
00:19:30.660 --> 00:19:34.530
another way to control
the bounded iterations.
00:19:34.530 --> 00:19:36.800
It says there is
some bounded set
00:19:36.800 --> 00:19:42.880
T such that this iterated
sum is nicely controlled.
00:19:42.880 --> 00:19:46.410
So let me iterate this
down even further.
00:19:46.410 --> 00:19:48.410
We're going to iterate
the containment
00:19:48.410 --> 00:19:58.360
by adding A to both sides.
00:19:58.360 --> 00:20:05.440
And we obtain that 3A minus
A is contained in T plus 2A
00:20:05.440 --> 00:20:14.370
minus A. But now, 2A minus A
was containing T plus A minus A.
00:20:14.370 --> 00:20:21.340
So we get 2T plus A minus A.
00:20:21.340 --> 00:20:27.150
So now we've gained
quite a bit, because we
00:20:27.150 --> 00:20:31.350
can make this iteration
go up, but not
00:20:31.350 --> 00:20:36.270
at the cost of iterating A but
at the cost of iterating T.
00:20:36.270 --> 00:20:39.060
But T is a bounded size.
00:20:39.060 --> 00:20:44.720
T is bounded size, so T
will have very nice control.
00:20:44.720 --> 00:20:48.210
We'll be able to very nicely
control the iterations of T.
00:20:48.210 --> 00:20:57.570
So if we keep going, we
find that for all integer n,
00:20:57.570 --> 00:21:02.780
positive integer n,
n plus 1 A minus A
00:21:02.780 --> 00:21:11.070
is contained in nT plus A minus
A. But for every integer n,
00:21:11.070 --> 00:21:14.640
the iterated sums
of T are contained
00:21:14.640 --> 00:21:27.740
in the subgroup generated by
T. So therefore, if we take n
00:21:27.740 --> 00:21:32.270
to be as large as you want,
see that the left-hand side
00:21:32.270 --> 00:21:36.670
eventually becomes the
subgroup generated by A,
00:21:36.670 --> 00:21:39.710
and the right hand side
does not depend on n.
00:21:42.590 --> 00:21:46.640
So you have this
containment over here.
00:21:46.640 --> 00:21:50.580
We would like to estimate how
large the subgroup generated
00:21:50.580 --> 00:21:52.450
by A is.
00:21:52.450 --> 00:21:55.100
So we look at that
formula, and we
00:21:55.100 --> 00:22:00.320
see that the size of the
subgroup generated by T--
00:22:00.320 --> 00:22:01.940
and here is where
we're going to use
00:22:01.940 --> 00:22:04.760
the fact that the assumption
that a group has bounded
00:22:04.760 --> 00:22:05.840
exponent.
00:22:05.840 --> 00:22:07.420
So think F2 to the
n, if you will.
00:22:07.420 --> 00:22:11.840
So in F2 to the n, if
I give you a set T,
00:22:11.840 --> 00:22:15.350
what's the subspace
spanned by T?
00:22:15.350 --> 00:22:18.390
What's the maximum
possible size?
00:22:18.390 --> 00:22:22.520
It's, at most, 2
raised to the number.
00:22:22.520 --> 00:22:25.630
So in general, you also
have that the subgroup
00:22:25.630 --> 00:22:31.060
generated by T has size at
most r raised to the size of T
00:22:31.060 --> 00:22:36.130
in a group of exponent
r, which we can control,
00:22:36.130 --> 00:22:39.370
because T has bounded size.
00:22:42.740 --> 00:22:48.650
And the second term,
A minus A, so we also
00:22:48.650 --> 00:22:51.192
know how to control
that by Plunnecke-Ruzsa.
00:23:00.850 --> 00:23:07.660
Therefore, putting
these two together,
00:23:07.660 --> 00:23:16.080
we see that the size up
here is at most r to the K
00:23:16.080 --> 00:23:22.002
to the 4 times K
squared size of A, which
00:23:22.002 --> 00:23:23.210
is the bound that we claimed.
00:23:28.320 --> 00:23:29.425
Any questions?
00:23:32.040 --> 00:23:36.640
So the trick here is to only
use Plunnecke-Ruzsa somehow
00:23:36.640 --> 00:23:38.110
a bounded number of times.
00:23:38.110 --> 00:23:43.000
If you use it too many
times, this bound blows up.
00:23:43.000 --> 00:23:45.250
But you use it only a
small number of times,
00:23:45.250 --> 00:23:49.590
and then you use the
Ruzsa covering lemma
00:23:49.590 --> 00:23:53.923
so that you get this bounded
set T that I can iterate.
00:23:57.630 --> 00:24:00.090
Let me make some comments
about the bounds that
00:24:00.090 --> 00:24:01.330
come out of this proof.
00:24:01.330 --> 00:24:05.850
So you get this very clean
bound, following this proof.
00:24:05.850 --> 00:24:07.870
But what about examples?
00:24:07.870 --> 00:24:09.390
So what kinds of
examples can you
00:24:09.390 --> 00:24:13.560
think of where you have a
set of bounded doubling,
00:24:13.560 --> 00:24:17.370
but the set generated, the
group generated by that set,
00:24:17.370 --> 00:24:20.100
is potentially large?
00:24:20.100 --> 00:24:34.570
So even in F2 to the n, so
if A is an independent set--
00:24:38.146 --> 00:24:41.360
so a basis or a subset of
a basis, for instance--
00:24:41.360 --> 00:24:47.820
then K-- so A is independent
set, so all the pairwise sums
00:24:47.820 --> 00:24:54.346
are distinct-- so K is
about the size of A/2.
00:24:54.346 --> 00:24:56.370
That's the doubling constant.
00:24:56.370 --> 00:25:02.710
Whereas, the group
generated by A
00:25:02.710 --> 00:25:11.720
has size 2 to the size of A,
which is around 2 to the 2K
00:25:11.720 --> 00:25:19.500
times the size of A. OK.
00:25:19.500 --> 00:25:24.210
So you see that you do need
some exponential blow-up from K
00:25:24.210 --> 00:25:27.810
to this constant over here.
00:25:27.810 --> 00:25:29.910
And turns out, that's
more or less correct.
00:25:29.910 --> 00:25:41.540
So the optimal constant
for F2 to the n
00:25:41.540 --> 00:25:44.060
is now known very precisely.
00:25:47.800 --> 00:25:53.310
And so if you give me a real
value of K, then I can tell you
00:25:53.310 --> 00:25:54.960
there are some
recent results that
00:25:54.960 --> 00:25:57.690
tells you exactly what is
the optimal constant you
00:25:57.690 --> 00:26:00.210
can put in front of
the A. So very precise.
00:26:00.210 --> 00:26:09.700
But asymptotically, it looks
like 2 to the 2K divided by.
00:26:09.700 --> 00:26:15.680
K So that's what it looks like.
00:26:15.680 --> 00:26:19.650
So this example is
basically correct.
00:26:19.650 --> 00:26:26.400
For general r, we expect
a similar phenomenon.
00:26:26.400 --> 00:26:31.635
So Ruzsa conjectured that--
00:26:31.635 --> 00:26:35.790
in the hypothesis
of the theorem,
00:26:35.790 --> 00:26:37.680
the constant you
can take is only
00:26:37.680 --> 00:26:42.930
exponential in K, the r to
the some constant C times K.
00:26:42.930 --> 00:26:48.840
And it has been verified
for some values of r,
00:26:48.840 --> 00:26:49.750
but not in general--
00:26:53.430 --> 00:26:55.215
for some r, for example primes.
00:27:02.625 --> 00:27:03.590
OK.
00:27:03.590 --> 00:27:04.583
Any questions?
00:27:09.650 --> 00:27:10.150
All right.
00:27:10.150 --> 00:27:13.420
So this is a good milestone.
00:27:13.420 --> 00:27:15.415
So we've developed
some tools, and we
00:27:15.415 --> 00:27:18.940
were able to prove a easier
version of Freiman's theorem
00:27:18.940 --> 00:27:22.630
in a group of bounded exponent.
00:27:22.630 --> 00:27:24.670
And you can ask
yourself, does this proof
00:27:24.670 --> 00:27:27.000
work in the integers?
00:27:27.000 --> 00:27:28.473
And well, literally no.
00:27:28.473 --> 00:27:29.890
Because if you
look at this proof,
00:27:29.890 --> 00:27:34.660
this set here is infinite,
unlike in the finite field
00:27:34.660 --> 00:27:35.760
setting.
00:27:35.760 --> 00:27:38.860
In the integers, well,
that's not very good.
00:27:38.860 --> 00:27:42.910
So the strategy of
Freiman's theorem,
00:27:42.910 --> 00:27:46.630
the proof of Freiman's theorem,
is to start with the integers,
00:27:46.630 --> 00:27:50.080
and then try to, not
work in the integers,
00:27:50.080 --> 00:27:52.840
but try to work in
a smaller group.
00:27:52.840 --> 00:27:55.690
Even though you start
in a maybe very spread
00:27:55.690 --> 00:27:59.260
out set of integers, I want to
work in a much smaller group
00:27:59.260 --> 00:28:03.390
so that I can control
things within that group.
00:28:03.390 --> 00:28:05.460
And this is an idea
called modeling.
00:28:05.460 --> 00:28:07.400
So I have a big set
and want to model it
00:28:07.400 --> 00:28:11.480
by something in a small group.
00:28:11.480 --> 00:28:12.890
So we're going to see this idea.
00:28:12.890 --> 00:28:16.040
But to understand
what does it mean
00:28:16.040 --> 00:28:18.470
to have a good model
for a set in the sense
00:28:18.470 --> 00:28:20.240
of additive
combinatorics, I need
00:28:20.240 --> 00:28:22.910
to introduce the notion
of Freiman homomorphisms.
00:28:40.120 --> 00:28:44.120
So one of the central
philosophies across mathematics
00:28:44.120 --> 00:28:46.610
is that if you want
to study objects,
00:28:46.610 --> 00:28:49.280
you should try to understand
maps between objects
00:28:49.280 --> 00:28:54.650
and understand properties that
are preserved under those maps.
00:28:54.650 --> 00:28:56.480
So if you want to
study groups, I
00:28:56.480 --> 00:28:58.740
don't really care how you
label your group elements--
00:28:58.740 --> 00:29:01.040
by 1, 2, 3, or A,
B, C. What I care
00:29:01.040 --> 00:29:02.690
about is the relationships.
00:29:02.690 --> 00:29:04.830
And those are the data
that I care about.
00:29:04.830 --> 00:29:06.680
And then, of course,
then you have
00:29:06.680 --> 00:29:08.720
concepts like group
homomorphisms,
00:29:08.720 --> 00:29:13.950
group isomorphisms, that
preserve all the relevant data.
00:29:13.950 --> 00:29:15.630
Similarly in any other area--
00:29:15.630 --> 00:29:17.240
in geometry you have manifolds.
00:29:17.240 --> 00:29:20.810
You understand not specifically
how they embed into space
00:29:20.810 --> 00:29:22.760
but what are the
intrinsic properties.
00:29:22.760 --> 00:29:24.290
So we would like
to understand what
00:29:24.290 --> 00:29:28.280
are the intrinsic properties
of a subset of an abelian group
00:29:28.280 --> 00:29:29.960
that we care about
for the purpose
00:29:29.960 --> 00:29:32.360
of additive combinatorics,
and specifically
00:29:32.360 --> 00:29:34.860
for Friedman's theorem.
00:29:34.860 --> 00:29:38.390
And what we care about is what
kinds of additive relationships
00:29:38.390 --> 00:29:40.160
are preserved.
00:29:40.160 --> 00:29:45.120
And Freiman's homomorphisms
capture that notion.
00:29:45.120 --> 00:29:46.880
So roughly speaking,
we would like
00:29:46.880 --> 00:29:52.810
to understand maps between sets
in possibly different groups--
00:29:57.500 --> 00:29:59.720
in possibly different
abelian groups--
00:30:03.390 --> 00:30:06.897
that only partially
preserve additive structure.
00:30:17.850 --> 00:30:21.840
So here's a definition.
00:30:21.840 --> 00:30:32.700
Suppose we have A
and B, and they're
00:30:32.700 --> 00:30:37.530
subsets in possibly
different abelian groups.
00:30:40.180 --> 00:30:42.740
Could be the same, but
possibly different.
00:30:42.740 --> 00:30:45.970
And everything's written
under addition, as usual.
00:30:45.970 --> 00:30:55.500
So we say that a
map phi from A to B
00:30:55.500 --> 00:31:03.010
is a Freiman s-homomorphism.
00:31:08.610 --> 00:31:09.620
So that's the term--
00:31:09.620 --> 00:31:13.750
Freiman s-homomorphism,
sometimes also Freiman
00:31:13.750 --> 00:31:20.380
homomorphisms of order s, so
equivalently I can call it
00:31:20.380 --> 00:31:22.000
that, as well--
00:31:22.000 --> 00:31:24.400
if the following holds.
00:31:24.400 --> 00:31:30.540
If we have the equation
phi of A plus dot,
00:31:30.540 --> 00:31:37.870
dot, dot plus phi of a sub
s equal to phi of a prime 1
00:31:37.870 --> 00:31:49.160
plus dot, dot, dot plus
phi of a prime s, whenever
00:31:49.160 --> 00:31:54.750
a through as, a prime
through as prime, so
00:31:54.750 --> 00:32:04.770
a1 prime through as prime,
satisfy the equation
00:32:04.770 --> 00:32:10.650
a1 plus dot, dot, dot plus as
equal to a1 prime plus dot,
00:32:10.650 --> 00:32:13.010
dot, dot plus as prime.
00:32:16.160 --> 00:32:17.330
OK.
00:32:17.330 --> 00:32:22.680
So that's the definition of
a Freiman s-homomorphism.
00:32:22.680 --> 00:32:27.110
It should remind you of
the definition of a group
00:32:27.110 --> 00:32:32.360
homomorphism, which completely
preserves additive structure,
00:32:32.360 --> 00:32:35.030
let's say, between
abelian groups.
00:32:35.030 --> 00:32:36.740
And for Freiman
homomorphisms, I'm
00:32:36.740 --> 00:32:40.292
only asking you to partially
preserve additive structure.
00:32:44.210 --> 00:32:46.670
So the point here is that
if I only care about,
00:32:46.670 --> 00:32:49.070
let's say, pairwise
sums, if that's
00:32:49.070 --> 00:32:52.720
the only data I care about,
then Freiman homomorphisms
00:32:52.720 --> 00:32:53.660
preserve that data.
00:33:01.830 --> 00:33:03.090
To give you some--
00:33:03.090 --> 00:33:04.580
OK, so one more thing.
00:33:08.720 --> 00:33:19.180
If phi from A to B is,
furthermore, a bijection,
00:33:19.180 --> 00:33:32.800
and both phi and phi inverse
are Freiman s-homomorphisms,
00:33:32.800 --> 00:33:39.940
then we say that phi is
a Freiman s-isomorphism.
00:33:48.850 --> 00:33:52.000
So it's not enough just to be a
bijection, but it's a bijection
00:33:52.000 --> 00:33:54.710
and both the forward
and the inverse
00:33:54.710 --> 00:33:59.970
maps are Freiman homomorphisms.
00:33:59.970 --> 00:34:03.240
So these are the definitions
we're going to use.
00:34:03.240 --> 00:34:04.490
Let me give you some examples.
00:34:09.510 --> 00:34:17.385
So every group homomorphism
is a Freiman homomorphism
00:34:17.385 --> 00:34:18.010
of every order.
00:34:24.677 --> 00:34:28.320
So group homomorphisms preserve
all additive structure,
00:34:28.320 --> 00:34:30.880
and Freiman homomorphisms
only partially
00:34:30.880 --> 00:34:34.500
preserve additive structure.
00:34:34.500 --> 00:34:40.560
A composition-- so
if phi 1 and phi 2
00:34:40.560 --> 00:34:49.020
are Freiman s-homomorphisms,
then phi 1 composed with phi 2
00:34:49.020 --> 00:34:53.764
is a Freiman s-homomorphism.
00:34:53.764 --> 00:34:56.900
So compositions
preserve this property.
00:34:56.900 --> 00:35:01.190
And likewise, instead
of homomorphisms,
00:35:01.190 --> 00:35:06.110
if you have isomorphisms, then
that's also true, as well.
00:35:06.110 --> 00:35:09.920
So these are straightforward
things to check.
00:35:09.920 --> 00:35:14.230
So a concrete example that shows
you a difference between group
00:35:14.230 --> 00:35:17.760
homomorphisms and
Freiman homomorphisms
00:35:17.760 --> 00:35:23.720
is, suppose you take
an arbitrary map
00:35:23.720 --> 00:35:29.270
phi from a set that has
no additive structure.
00:35:29.270 --> 00:35:35.570
So it's a four-element set,
has no additive structure.
00:35:35.570 --> 00:35:39.590
And I map it to
the integers, claim
00:35:39.590 --> 00:35:43.850
that this is a Freiman
2-homomorphism.
00:35:49.790 --> 00:35:50.620
So you can check.
00:35:50.620 --> 00:35:54.010
So whenever this is
satisfied, but that's
00:35:54.010 --> 00:35:56.730
never non-trivially satisfied.
00:35:56.730 --> 00:36:00.078
So an arbitrary map here is
a Freiman 2-homomorphism.
00:36:03.070 --> 00:36:08.560
And if furthermore--
so if you have,
00:36:08.560 --> 00:36:13.670
let's say, bijection
between two sets,
00:36:13.670 --> 00:36:27.770
both having no additive
structure, if it's a bijection,
00:36:27.770 --> 00:36:32.390
it's a Freiman isomorphism
of here, order 2.
00:36:36.960 --> 00:36:38.460
Let me give you a
few more examples.
00:36:53.450 --> 00:36:58.190
When you look at homomorphisms
between finite groups,
00:36:58.190 --> 00:37:01.670
so you know that if
you have a homomorphism
00:37:01.670 --> 00:37:08.470
and it's also a bijection,
then it's an isomorphism.
00:37:08.470 --> 00:37:13.450
But that's not true for this
notion of homomorphisms.
00:37:25.190 --> 00:37:34.110
So the natural embedding
that sends the Boolean cube
00:37:34.110 --> 00:37:39.590
to the Boolean cube viewed
as Z mod 2 to the n.
00:37:45.990 --> 00:37:47.700
So what's happening here?
00:37:50.790 --> 00:37:53.600
This is a part of a
group homomorphism.
00:37:53.600 --> 00:37:56.870
And so if you look at Z
to the n, and I do mod 2,
00:37:56.870 --> 00:38:00.650
and I restrict to
this Boolean cube,
00:38:00.650 --> 00:38:03.950
that's the group
homomorphism If I view this
00:38:03.950 --> 00:38:05.870
as a subset of a bigger group.
00:38:05.870 --> 00:38:11.150
So it is a Freiman
homomorphisms of every order.
00:38:15.350 --> 00:38:17.510
And it's bijective.
00:38:17.510 --> 00:38:22.730
But it is not a
Freiman 2-isomorphism.
00:38:27.570 --> 00:38:30.240
Because you have
additive relations here
00:38:30.240 --> 00:38:32.890
that are not present over here.
00:38:32.890 --> 00:38:36.000
So if you read the
definition, the inverse map,
00:38:36.000 --> 00:38:37.950
there are some
additive relations here
00:38:37.950 --> 00:38:39.660
that are not preserved
if you pull back.
00:38:45.310 --> 00:38:46.960
Here's another
example that will be
00:38:46.960 --> 00:38:49.330
more relevant to our
subsequent discussion.
00:38:53.270 --> 00:39:01.170
So the mod N map, which sends Z
to Z mod N, so this is a group
00:39:01.170 --> 00:39:04.060
homomorphisms.
00:39:04.060 --> 00:39:09.580
So hence, it's a Freiman
homomorphism of every order.
00:39:09.580 --> 00:39:11.840
But it's not--
00:39:11.840 --> 00:39:20.050
OK, so if you look at this map,
and even if I restrict to here,
00:39:20.050 --> 00:39:26.720
so it's not a Freiman
isomorphism just like earlier.
00:39:26.720 --> 00:39:39.560
However-- let me
go back to Z. So
00:39:39.560 --> 00:39:51.000
if A is a subset of integers
with diameter less than N/s,
00:39:51.000 --> 00:40:05.900
then this map mod N maps
A Freiman s-isomorphically
00:40:05.900 --> 00:40:06.770
onto its image.
00:40:12.810 --> 00:40:16.310
So even though mod N
restricted to 1 through N
00:40:16.310 --> 00:40:21.170
is not a Freiman
isomorphism of order 2,
00:40:21.170 --> 00:40:26.130
if I restrict to a
subset that's, let's say,
00:40:26.130 --> 00:40:29.130
contained in some
small interval,
00:40:29.130 --> 00:40:34.340
then all the additive
structures are preserved.
00:40:34.340 --> 00:40:35.420
So let me show you why.
00:40:35.420 --> 00:40:37.910
So this is not too hard
once you get your head
00:40:37.910 --> 00:40:40.740
around the definition.
00:40:40.740 --> 00:40:52.250
So indeed, if you have group
elements a1 through as,
00:40:52.250 --> 00:41:02.777
a prime 1 through a prime s, and
if they satisfy the equation--
00:41:13.990 --> 00:41:15.490
so if they satisfy
this equations,
00:41:15.490 --> 00:41:20.410
so we're trying to verify that
it is a Freiman s-isomorphism,
00:41:20.410 --> 00:41:24.350
namely the inverse of this map
is a Freiman s-homomorphism,
00:41:24.350 --> 00:41:27.920
so if they satisfy
this equation,
00:41:27.920 --> 00:41:32.050
so this is satisfying this
additive relation in the image,
00:41:32.050 --> 00:41:40.560
in Z mod N, then note
that the left-hand side--
00:41:40.560 --> 00:41:43.410
so all of these A's are
contained in a small interval,
00:41:43.410 --> 00:41:46.690
because the diameter of
the set is less than N/s.
00:41:46.690 --> 00:41:52.950
So if you look at how big a1
minus a1 prime can be it's,
00:41:52.950 --> 00:41:56.820
at most-- or, it's
less than N/s in size.
00:41:56.820 --> 00:41:58.970
So the left-hand side,
in absolute value,
00:41:58.970 --> 00:41:59.910
viewed as an integer--
00:42:03.660 --> 00:42:12.510
so the left-hand side is less
than N in absolute value,
00:42:12.510 --> 00:42:18.940
since the diameter of
a is less than N/s.
00:42:18.940 --> 00:42:22.945
So you have some number here,
which is strictly less than N
00:42:22.945 --> 00:42:27.850
in absolute value,
and it's 0 mod N,
00:42:27.850 --> 00:42:35.230
so it must be actually equal
to 0 as a number as an integer.
00:42:35.230 --> 00:42:38.790
So this verifies that
the additive relations
00:42:38.790 --> 00:42:43.590
up to s-wise sums are
preserved under the mod N map,
00:42:43.590 --> 00:42:46.590
if you restrict to
a small interval.
00:42:54.715 --> 00:42:55.590
Any questions so far?
00:42:58.200 --> 00:43:00.970
So in additive
combinatorics, we are
00:43:00.970 --> 00:43:05.980
trying to understand properties,
specific additive properties.
00:43:05.980 --> 00:43:08.560
And the notion of
Freiman homomorphisms
00:43:08.560 --> 00:43:12.490
Freiman isomorphism capture
what specific properties we
00:43:12.490 --> 00:43:14.530
need to study and
what are the maps that
00:43:14.530 --> 00:43:17.620
preserve those properties.
00:43:17.620 --> 00:43:21.280
And the next thing we will do is
to understand this model lemma,
00:43:21.280 --> 00:43:22.990
the modeling lemma,
that tells us
00:43:22.990 --> 00:43:27.620
that if you start with a set A
with small doubling, initially
00:43:27.620 --> 00:43:30.550
A may be very much spread
out in the integers-- it may
00:43:30.550 --> 00:43:33.140
have very large elements, very
small elements, very spread
00:43:33.140 --> 00:43:33.640
out.
00:43:33.640 --> 00:43:36.040
But if A small
doubling properties,
00:43:36.040 --> 00:43:42.880
then I can model A inside
a small group, such
00:43:42.880 --> 00:43:46.860
that all the
relevant data, namely
00:43:46.860 --> 00:43:49.170
relative to these
Freiman homomorphisms,
00:43:49.170 --> 00:43:50.780
are preserved under this model.
00:43:53.550 --> 00:43:55.200
So let's move on to
the modeling lemma.
00:44:06.180 --> 00:44:09.540
The main message of
the modeling lemma
00:44:09.540 --> 00:44:26.440
is that if A has small doubling,
then A can be modeled--
00:44:26.440 --> 00:44:30.700
and here, that means
being Freiman isomorphic--
00:44:30.700 --> 00:44:37.400
to a subset of a small group.
00:44:43.990 --> 00:44:49.100
So first as a warm-up, let's
work in the finite field model,
00:44:49.100 --> 00:44:51.610
just to see what such
a result looks like.
00:44:51.610 --> 00:44:53.420
And it contains
most of the ideas,
00:44:53.420 --> 00:44:54.710
but it's much more clean.
00:44:54.710 --> 00:44:57.230
It's much cleaner
than in the integers.
00:44:57.230 --> 00:45:06.382
So in the finite field model,
specifically F2 to the n,
00:45:06.382 --> 00:45:07.340
what do we want to say?
00:45:11.720 --> 00:45:16.340
Suppose you have A, a
subset of F2 to the n,
00:45:16.340 --> 00:45:23.000
and suppose that m is some
number such that 2 to the m
00:45:23.000 --> 00:45:28.580
is at least as large
as sA minus sA.
00:45:28.580 --> 00:45:31.040
So remember, from
Plunnecke-Ruzsa,
00:45:31.040 --> 00:45:34.380
you know that if
A plus A is small,
00:45:34.380 --> 00:45:35.990
then this iterated sum is small.
00:45:38.500 --> 00:45:42.780
So suppose we have these
parameters and sets.
00:45:42.780 --> 00:45:50.770
The conclusion is
that A is Freiman
00:45:50.770 --> 00:46:06.900
s-isomorphic to some
subset of F2 raised to m.
00:46:06.900 --> 00:46:10.650
So initially, A is in a
potentially very large vector
00:46:10.650 --> 00:46:13.960
space, or it could be
all over the place.
00:46:13.960 --> 00:46:16.650
And what we are
trying to say here
00:46:16.650 --> 00:46:31.740
is that if A has small doubling,
then by Plunnecke-Ruzsa, sA
00:46:31.740 --> 00:46:36.130
minus sA has size not too much
bigger than A itself-- only
00:46:36.130 --> 00:46:38.640
bounded times the
size of A itself.
00:46:38.640 --> 00:46:42.840
So I can take an m so that
the size of this group
00:46:42.840 --> 00:46:45.600
is only a constant
factor in larger
00:46:45.600 --> 00:46:46.950
than the size of A itself.
00:46:50.780 --> 00:46:52.460
So we're in a
pretty small group.
00:46:52.460 --> 00:46:56.620
So we are able to model A,
even though initially it
00:46:56.620 --> 00:46:58.430
sits inside a pretty
large abelian group,
00:46:58.430 --> 00:47:01.270
by some subset in a
pretty small group.
00:47:03.593 --> 00:47:05.510
So let's see how to prove
this modeling lemma.
00:47:10.450 --> 00:47:18.290
So the finite field
setting is much easier to--
00:47:18.290 --> 00:47:21.200
it's not too hard to
deal with, because we
00:47:21.200 --> 00:47:22.940
can look at linear maps.
00:47:22.940 --> 00:47:33.960
So the following are
equivalent for linear maps phi,
00:47:33.960 --> 00:47:37.610
so for group homomorphisms.
00:47:37.610 --> 00:47:43.240
So phi is a Freiman
s-isomorphism
00:47:43.240 --> 00:47:49.770
when restricted
to A. So here, phi
00:47:49.770 --> 00:47:56.450
I'm going to let it be a linear
map from F2 to the n to F2
00:47:56.450 --> 00:47:58.030
to the m.
00:47:58.030 --> 00:48:02.710
The following are equivalent.
00:48:02.710 --> 00:48:07.020
So we would like phi to be
a Freiman s-isomorphism when
00:48:07.020 --> 00:48:10.830
restricted to A. Because this
means that when I restrict to A
00:48:10.830 --> 00:48:14.670
and I restrict to its image,
it Freiman isomorphically maps
00:48:14.670 --> 00:48:17.312
onto the image.
00:48:17.312 --> 00:48:18.270
So what does that mean?
00:48:22.060 --> 00:48:23.830
So phi is already
a homomorphism.
00:48:23.830 --> 00:48:27.310
So it's automatically a
Freiman s-homomorphism.
00:48:27.310 --> 00:48:30.490
For it to be an
s-isomorphism, it just
00:48:30.490 --> 00:48:33.610
means that there are no
additional linear relations
00:48:33.610 --> 00:48:38.660
in the image that were not
present earlier, which means
00:48:38.660 --> 00:48:45.390
that phi is injective on sA.
00:48:48.600 --> 00:48:51.490
So let's just think
about the definition.
00:48:51.490 --> 00:48:54.715
And in the definition,
if you know additionally
00:48:54.715 --> 00:48:59.230
that phi is a homomorphism,
everything's much cleaner.
00:48:59.230 --> 00:49:02.740
It is also equivalent
to that phi
00:49:02.740 --> 00:49:10.180
of x is non-zero for
every non-zero element
00:49:10.180 --> 00:49:13.210
x of sA minus sA.
00:49:17.470 --> 00:49:19.780
So this is a very
clean characterization
00:49:19.780 --> 00:49:22.810
of what it means to be
in Freiman s-isomorphism
00:49:22.810 --> 00:49:25.810
when you are in an
abelian group and you have
00:49:25.810 --> 00:49:28.750
linear maps of homomorphisms.
00:49:28.750 --> 00:49:33.520
So if we start by
taking phi to be
00:49:33.520 --> 00:49:41.560
a uniformly random linear map--
00:49:46.920 --> 00:49:49.420
so for example,
you pick a basis,
00:49:49.420 --> 00:49:52.420
and you send each basis element
to a uniformly random element--
00:49:55.720 --> 00:50:04.920
then we find that if 2m is at
least A at minus sA, then--
00:50:04.920 --> 00:50:08.150
so let me call these
properties 1, 2, 3--
00:50:08.150 --> 00:50:17.010
so then the probability that
3 is satisfied is positive.
00:50:17.010 --> 00:50:22.610
Because each element
of sA minus sA--
00:50:22.610 --> 00:50:28.060
I can also ignore 0-- so each
non-zero element of sA minus sA
00:50:28.060 --> 00:50:32.260
violates this property
with probability exactly 2
00:50:32.260 --> 00:50:36.930
to the minus m,
everything is uniform.
00:50:36.930 --> 00:50:39.820
So if there are
very few elements,
00:50:39.820 --> 00:50:44.620
and the space is large
enough, then the third bullet
00:50:44.620 --> 00:50:47.360
is satisfied with
positive probability.
00:50:47.360 --> 00:50:50.260
So you get Freiman
s-isomorphism.
00:50:53.640 --> 00:50:54.680
Any questions?
00:50:57.440 --> 00:51:00.890
To get this model, in this case
in the finite field setting,
00:51:00.890 --> 00:51:01.610
it's not so hard.
00:51:01.610 --> 00:51:03.650
So you kind of
project the whole set,
00:51:03.650 --> 00:51:07.370
even though initially it might
have or be very spread out
00:51:07.370 --> 00:51:09.170
into a lot of dimensions.
00:51:09.170 --> 00:51:15.040
You project it down to a small
dimensional subspace randomly,
00:51:15.040 --> 00:51:17.050
and that works.
00:51:17.050 --> 00:51:18.860
So then with high
probability, it
00:51:18.860 --> 00:51:21.360
preserves all the additive
structures that you want,
00:51:21.360 --> 00:51:24.890
provided that you
have small doubling.
00:51:24.890 --> 00:51:34.210
Now, let's look at
what happens in Z.
00:51:34.210 --> 00:51:38.690
So in Z, things are
a bit more involved.
00:51:38.690 --> 00:51:41.180
But the ideas-- actually,
a lot of the ideas--
00:51:41.180 --> 00:51:43.490
come from this proof, as well.
00:51:43.490 --> 00:51:48.260
So Ruzsa's modeling
lemma tells us
00:51:48.260 --> 00:51:53.880
that if you have a
set of integers--
00:51:53.880 --> 00:51:55.650
always a finite set--
00:51:55.650 --> 00:52:11.790
and integers s and N are such
that N is at least sA minus sA,
00:52:11.790 --> 00:52:13.770
then so it turns
out you might not
00:52:13.770 --> 00:52:16.350
be able to model
the whole set A.
00:52:16.350 --> 00:52:20.660
But it will be good enough for
us to model a large fraction.
00:52:20.660 --> 00:52:28.000
So then there exists
an A prime subset of A,
00:52:28.000 --> 00:52:36.930
with A prime being at least an
s fraction of the original set.
00:52:36.930 --> 00:52:48.130
And A prime is Freiman
s-isomorphic to a subset
00:52:48.130 --> 00:52:55.940
of Z mod N.
00:52:55.940 --> 00:53:03.490
So same message as before,
with an extra ingredient
00:53:03.490 --> 00:53:05.795
that we did not see before.
00:53:05.795 --> 00:53:07.170
But the point is
that if you have
00:53:07.170 --> 00:53:11.320
a set A with controlled
doubling, then well, now
00:53:11.320 --> 00:53:14.830
you can take a large
fraction of A that
00:53:14.830 --> 00:53:20.260
is Freiman isomorphism to
a subset of a small group.
00:53:20.260 --> 00:53:22.700
This is small,
because we only need
00:53:22.700 --> 00:53:28.660
n to exceed sA minus sA, which
are only constant factor more
00:53:28.660 --> 00:53:31.414
than the size of A. Yeah?
00:53:31.414 --> 00:53:33.850
AUDIENCE: Is s greater
than 2 comma N?
00:53:33.850 --> 00:53:34.600
YUFEI ZHAO: Sorry.
00:53:34.600 --> 00:53:40.960
S greater than 2 separate
and some integer.
00:53:40.960 --> 00:53:43.240
Thank you.
00:53:43.240 --> 00:53:45.930
So in our application,
s will be a constant.
00:53:45.930 --> 00:53:48.690
s will be 8.
00:53:48.690 --> 00:53:50.895
So think of s as some
specific constant.
00:53:54.210 --> 00:53:55.382
Any questions?
00:53:57.980 --> 00:54:01.910
So let's prove this
modeling lemma.
00:54:01.910 --> 00:54:07.930
We want to try to do
some kind of random map.
00:54:07.930 --> 00:54:12.460
But it's not clear how to
start doing a random map
00:54:12.460 --> 00:54:15.740
if you just start
in the integers.
00:54:15.740 --> 00:54:19.480
So what we want to do
is first place ourself
00:54:19.480 --> 00:54:27.520
in some group where we can
consider random automorphisms.
00:54:27.520 --> 00:54:29.800
So we start by--
00:54:29.800 --> 00:54:35.860
perhaps we're very wastefully
choosing a prime q bigger
00:54:35.860 --> 00:54:40.240
than the maximum
possible sA minus sA.
00:54:43.840 --> 00:54:47.080
And so just choose a
large enough prime.
00:54:47.080 --> 00:54:48.610
I don't care how large you pick.
00:54:48.610 --> 00:54:51.520
q can be very, very large.
00:54:51.520 --> 00:54:52.570
Pick a prime.
00:54:52.570 --> 00:54:57.520
And now I work inside Z mod q.
00:54:57.520 --> 00:55:00.230
I noticed that if you
make q large enough,
00:55:00.230 --> 00:55:03.130
then A sits inside
Z mod q Freiman
00:55:03.130 --> 00:55:05.820
isomorphically, or
s-isomorphically.
00:55:05.820 --> 00:55:08.350
So just pick q large
enough so that you don't
00:55:08.350 --> 00:55:11.580
have to worry about any issues.
00:55:11.580 --> 00:55:13.870
So Yeah.
00:55:13.870 --> 00:55:20.810
So the mod q map
from A to Z mod q--
00:55:20.810 --> 00:55:26.790
so this is Freiman
s-isomorphic--
00:55:26.790 --> 00:55:28.320
onto its image.
00:55:32.400 --> 00:55:36.968
So let's now consider
a sequence of maps.
00:55:36.968 --> 00:55:39.010
And we're going to denote
the sequence like this.
00:55:39.010 --> 00:55:43.550
So we start with Z. That's
where A originally sits.
00:55:43.550 --> 00:55:51.060
And maps to Z mod q-- so that
was the first map that we saw.
00:55:51.060 --> 00:55:54.150
And now we want to do a
random automorphism, kind
00:55:54.150 --> 00:55:55.770
of like the random map earlier.
00:55:55.770 --> 00:56:00.680
But in Z mod q, there are lots
of nice random automorphisms,
00:56:00.680 --> 00:56:06.940
namely multiplication by
some non-zero element.
00:56:06.940 --> 00:56:12.460
And finally, we can consider
the representative map,
00:56:12.460 --> 00:56:16.150
where every element of
Z mod q, I can associate
00:56:16.150 --> 00:56:21.080
to it a positive
integer from 1 to q
00:56:21.080 --> 00:56:22.820
which agrees with the Z mod q.
00:56:22.820 --> 00:56:26.890
So the final step is not
a group homomorphism.
00:56:26.890 --> 00:56:29.870
So we need to be more careful.
00:56:29.870 --> 00:56:32.760
So let me denote by
phi this entire map.
00:56:32.760 --> 00:56:34.950
So from the beginning to
the end, this composition
00:56:34.950 --> 00:56:38.000
I'll denote by phi.
00:56:38.000 --> 00:56:44.340
And lambda here is some element
between 1 and q minus 1.
00:56:47.000 --> 00:56:49.540
Now, remember what
we said earlier,
00:56:49.540 --> 00:56:52.930
that this map, this
final map here,
00:56:52.930 --> 00:56:56.680
might not be a Freiman
homomorphism, because there
00:56:56.680 --> 00:57:01.630
are some additive relations
here that are not preserved over
00:57:01.630 --> 00:57:02.650
here.
00:57:02.650 --> 00:57:07.360
But if I restrict myself
to a small interval,
00:57:07.360 --> 00:57:10.300
then it is a
Freiman homomorphism
00:57:10.300 --> 00:57:13.290
if we restrict to that interval.
00:57:13.290 --> 00:57:15.060
If you restrict
yourself to an interval,
00:57:15.060 --> 00:57:18.330
you cannot have extra
relations over here,
00:57:18.330 --> 00:57:19.395
because they cannot--
00:57:19.395 --> 00:57:22.470
the interval is small enough,
you can't wrap around.
00:57:22.470 --> 00:57:27.330
So let's consider restrictions
to small intervals.
00:57:27.330 --> 00:57:29.190
I start with A over here.
00:57:29.190 --> 00:57:32.640
So I want to restrict
myself to some interval
00:57:32.640 --> 00:57:38.670
so that I still have lots
of A in that restriction.
00:57:38.670 --> 00:57:41.100
And you can do
this by pigeonhole.
00:57:41.100 --> 00:57:46.710
So by pigeonhole,
for every lambda
00:57:46.710 --> 00:57:52.670
there exists some
interval we'll denote
00:57:52.670 --> 00:57:58.560
by I sub lambda inside q.
00:57:58.560 --> 00:58:05.250
So the length of this
interval will be at most q/s,
00:58:05.250 --> 00:58:10.940
such that if I look at the
restriction of this interval,
00:58:10.940 --> 00:58:16.550
I pull it all the way back to
the beginning then I still get
00:58:16.550 --> 00:58:20.950
a lot of elements of A.
So A sub lambda, namely
00:58:20.950 --> 00:58:29.570
the elements of A whose map
gets sent to this interval,
00:58:29.570 --> 00:58:33.820
has at least A/s elements.
00:58:37.180 --> 00:58:41.780
So for instance, you can chop
up q into s different intervals.
00:58:41.780 --> 00:58:51.712
So one of them will have lots
of elements that came from A.
00:58:51.712 --> 00:58:54.310
And this is why in
the end we only get
00:58:54.310 --> 00:58:56.410
a large subset of
the A. So we're
00:58:56.410 --> 00:58:58.240
going to forget
about everything else
00:58:58.240 --> 00:59:01.078
and focus our attention
on the set here.
00:59:05.760 --> 00:59:09.990
So thus, by our
earlier discussion
00:59:09.990 --> 00:59:16.020
having to do with the final map
being a Freiman s-homomorphism
00:59:16.020 --> 00:59:19.440
when you're working
inside a short interval,
00:59:19.440 --> 00:59:25.380
we see that phi, if you
restrict to this A sub lambda,
00:59:25.380 --> 00:59:30.240
is a Freiman s-homomorphism.
00:59:32.760 --> 00:59:34.710
Each step is a
Freiman homomorphism,
00:59:34.710 --> 00:59:36.540
because it's a
group homomorphism.
00:59:36.540 --> 00:59:39.900
And the final step
in the restriction
00:59:39.900 --> 00:59:43.440
is also a Freiman
s-homomorphism,
00:59:43.440 --> 00:59:49.710
because what we said about
working inside short intervals.
00:59:49.710 --> 00:59:51.050
So this part is very good.
00:59:53.870 --> 00:59:54.470
All right.
00:59:54.470 --> 00:59:57.000
So now let me consider
one more composition.
00:59:57.000 --> 00:59:58.520
So at the end of
the day, we would
00:59:58.520 --> 01:00:08.150
like to model this A lambda
inside some small cyclic group.
01:00:08.150 --> 01:00:11.390
So far, we don't have
small cyclic groups here.
01:00:11.390 --> 01:00:13.970
But I'm going to manufacture
a small cyclic group.
01:00:13.970 --> 01:00:20.780
So we're going to consider
the map where, first we
01:00:20.780 --> 01:00:23.960
take our phi all the
way until the end,
01:00:23.960 --> 01:00:28.850
and now you take mod m map.
01:00:28.850 --> 01:00:31.450
So if I don't write anything,
if it goes to Z mod m,
01:00:31.450 --> 01:00:34.230
it means the mod m map.
01:00:34.230 --> 01:00:39.470
So let me consider psi, which
is the composition of these two
01:00:39.470 --> 01:00:39.970
maps.
01:00:44.910 --> 01:00:46.600
All right.
01:00:46.600 --> 01:00:52.720
So we would like to say that
you can choose this lambda so
01:00:52.720 --> 01:00:59.020
that this A sub lambda gets
mapped Freiman s-isomorphically
01:00:59.020 --> 01:01:02.480
until to the end.
01:01:02.480 --> 01:01:04.130
So far, everything
looks pretty good.
01:01:04.130 --> 01:01:07.420
So you have Freiman
s-homomorphism,
01:01:07.420 --> 01:01:09.640
and you have a
group homomorphism.
01:01:09.640 --> 01:01:12.231
So the whole thing is a
Freiman s-homomorphism.
01:01:14.940 --> 01:01:20.250
So psi is restricted
to A sub lambda,
01:01:20.250 --> 01:01:23.226
is a Freiman s-homomorphism.
01:01:28.280 --> 01:01:31.130
But now the thing that
we really want to check
01:01:31.130 --> 01:01:37.160
is if there are some
relationships that
01:01:37.160 --> 01:01:42.950
are present at the
end in Z mod m that
01:01:42.950 --> 01:01:44.770
were not present earlier.
01:01:49.790 --> 01:01:52.470
And so we need to check that--
01:01:52.470 --> 01:02:03.320
we claim that if psi
does not map A sub lambda
01:02:03.320 --> 01:02:06.530
Freiman isomorphically,
then something
01:02:06.530 --> 01:02:08.010
has to have gone wrong.
01:02:08.010 --> 01:02:12.070
So if it does not map
A sub lambda Freiman
01:02:12.070 --> 01:02:19.430
isomorphically onto
its image, then
01:02:19.430 --> 01:02:21.560
what could have gone wrong?
01:02:21.560 --> 01:02:26.630
Claim that there
must be some d which
01:02:26.630 --> 01:02:34.070
depends on lambda
in sA minus sA,
01:02:34.070 --> 01:02:45.055
and d not 0, such that
phi of d is 0 mod m.
01:02:49.510 --> 01:02:50.560
So we'll prove this.
01:02:50.560 --> 01:02:53.000
But like before, it's
a very similar idea
01:02:53.000 --> 01:02:54.530
to what's happening earlier.
01:02:54.530 --> 01:02:56.990
The idea is that if you have--
01:02:59.600 --> 01:03:02.190
we want to show that there
are no additional additive
01:03:02.190 --> 01:03:05.710
relations in the image.
01:03:05.710 --> 01:03:07.370
So we would like--
01:03:07.370 --> 01:03:09.840
so if it's a
Freiman isomorphism,
01:03:09.840 --> 01:03:12.330
then there has to be some
accidental collisions.
01:03:12.330 --> 01:03:18.230
And that accidental collision
has to be witnessed by some d.
01:03:18.230 --> 01:03:19.780
So this requires some checking.
01:03:24.130 --> 01:03:33.360
So suppose-- indeed,
suppose the hypothesis--
01:03:33.360 --> 01:03:38.010
suppose that the psi does
not map A sub lambda Freiman
01:03:38.010 --> 01:03:41.130
s-isomorphically onto its image.
01:03:41.130 --> 01:03:51.720
Then there exists a1 through
as, a1 prime through as prime,
01:03:51.720 --> 01:04:03.740
in A lambda such that they do
not have additive relation,
01:04:03.740 --> 01:04:08.700
but their images do have
this additive relation.
01:04:12.900 --> 01:04:14.740
Their images all the
way until the end
01:04:14.740 --> 01:04:16.510
having the additive
relation means
01:04:16.510 --> 01:04:21.610
that phi has this
additive relation mod m.
01:04:35.030 --> 01:04:35.530
OK.
01:04:35.530 --> 01:04:36.322
So how can this be?
01:04:39.490 --> 01:04:49.740
Recall that since the image--
01:04:49.740 --> 01:04:52.610
so all of these elements-- lie
inside some short interval.
01:04:58.526 --> 01:05:05.400
The interval has length
less than 2 minus s.
01:05:05.400 --> 01:05:09.270
So we saw this argument, very
similar argument earlier,
01:05:09.270 --> 01:05:11.220
before the break.
01:05:11.220 --> 01:05:13.830
Because everything lies
in the short interval,
01:05:13.830 --> 01:05:18.150
we see that this difference
between the left-
01:05:18.150 --> 01:05:31.440
and the right-hand
sides, this difference
01:05:31.440 --> 01:05:33.670
is strictly less than q.
01:05:39.610 --> 01:05:49.020
Now, by switching the a's
and a primes if necessary,
01:05:49.020 --> 01:05:58.120
we may assume that this
difference is non-negative.
01:05:58.120 --> 01:06:00.460
Otherwise, it's just
a labeling issue.
01:06:00.460 --> 01:06:01.590
Otherwise, I relabel them.
01:06:05.280 --> 01:06:07.560
So then this here--
01:06:07.560 --> 01:06:11.510
so what's inside
this expression--
01:06:11.510 --> 01:06:16.442
we call this expression
inside the absolute value,
01:06:16.442 --> 01:06:17.520
we call it star.
01:06:22.110 --> 01:06:29.450
So star is some number between
0 and strictly-- so at least 0
01:06:29.450 --> 01:06:30.770
and strictly less than q.
01:06:39.560 --> 01:06:40.510
Right.
01:06:40.510 --> 01:06:57.950
So if we set d to be
this expression, so
01:06:57.950 --> 01:07:03.580
the difference between these two
sums, on one hand this d here--
01:07:06.350 --> 01:07:09.150
sorry, that's what
I want to say.
01:07:09.150 --> 01:07:10.180
Suppose you don't have--
01:07:16.420 --> 01:07:20.640
if you are not mapping Freiman
isomorphically onto the image,
01:07:20.640 --> 01:07:24.300
then I can exhibit some witness
for that non-isomorphism,
01:07:24.300 --> 01:07:27.780
meaning a bunch of
elements that do not
01:07:27.780 --> 01:07:30.600
have additive relations
in the domain.
01:07:30.600 --> 01:07:33.970
But I do have additive
relation in image.
01:07:33.970 --> 01:07:40.290
So if we set this d, then it's
some element of sA minus sA.
01:07:40.290 --> 01:07:44.760
And it's non-zero, because
we assume that d is non-zero.
01:07:50.860 --> 01:08:06.950
And so then, what can
we say about phi of d?
01:08:06.950 --> 01:08:16.850
So phi of d, I claim,
must be this expression
01:08:16.850 --> 01:08:23.399
over here, the difference
of the corresponding sums
01:08:23.399 --> 01:08:24.920
in the image.
01:08:24.920 --> 01:08:35.569
Because the two sides
are congruent mod q--
01:08:39.890 --> 01:08:42.560
two sides are congruent mod q.
01:08:42.560 --> 01:08:54.680
And furthermore, they are
in the interval from 0
01:08:54.680 --> 01:08:56.558
to strictly less than q.
01:09:09.529 --> 01:09:12.300
So this is a slightly
subtle argument.
01:09:12.300 --> 01:09:14.430
But the idea is all very simple.
01:09:14.430 --> 01:09:18.043
Just have to keep track
of the relationships
01:09:18.043 --> 01:09:19.960
between what's happening
in the domain, what's
01:09:19.960 --> 01:09:20.918
happening in the image.
01:09:23.295 --> 01:09:24.670
Somehow, I think
the finite field
01:09:24.670 --> 01:09:27.910
case is quite illustrative of--
01:09:27.910 --> 01:09:30.062
there, what goes wrong
is similar to what
01:09:30.062 --> 01:09:30.729
goes wrong here.
01:09:30.729 --> 01:09:32.979
Except here, you have to
keep track a bit more things.
01:09:35.840 --> 01:09:36.340
OK.
01:09:36.340 --> 01:09:42.430
So consequently,
thus phi of d is
01:09:42.430 --> 01:09:50.987
congruent to 0 mod m, which
is what we're looking for.
01:09:50.987 --> 01:09:52.029
So that proves the claim.
01:09:54.570 --> 01:09:55.584
Any questions?
01:09:58.902 --> 01:09:59.850
Yeah?
01:09:59.850 --> 01:10:01.225
AUDIENCE: [INAUDIBLE]
01:10:01.225 --> 01:10:01.850
YUFEI ZHAO: OK.
01:10:01.850 --> 01:10:03.060
So this part?
01:10:03.060 --> 01:10:04.160
All right.
01:10:04.160 --> 01:10:07.760
So we set d to be
this expression.
01:10:07.760 --> 01:10:10.070
I claim this equality.
01:10:10.070 --> 01:10:11.950
So why is it true?
01:10:11.950 --> 01:10:18.600
First, the left-hand side
and the right-hand side, they
01:10:18.600 --> 01:10:22.625
are congruent to
each other mod q.
01:10:22.625 --> 01:10:23.250
So why is that?
01:10:31.186 --> 01:10:33.412
AUDIENCE: [INAUDIBLE]
01:10:33.412 --> 01:10:34.162
YUFEI ZHAO: Sorry?
01:10:41.106 --> 01:10:43.090
Yeah.
01:10:43.090 --> 01:10:47.140
AUDIENCE: [INAUDIBLE] every
part of phi should preserve--
01:10:47.140 --> 01:10:49.740
the first two parts are
group homomorphisms.
01:10:49.740 --> 01:10:51.580
And then we just take
[INAUDIBLE] mod q.
01:10:51.580 --> 01:10:52.413
YUFEI ZHAO: Exactly.
01:10:52.413 --> 01:10:54.330
If you look at phi--
01:10:54.330 --> 01:11:00.180
where was it?-- up there, you
see that everything preserves.
01:11:00.180 --> 01:11:04.040
Even though the very last step
is not a group homomorphism,
01:11:04.040 --> 01:11:07.260
mod q is preserved.
01:11:07.260 --> 01:11:10.210
So even though the last step
is not group homomorphism,
01:11:10.210 --> 01:11:12.012
it is all mod q.
01:11:12.012 --> 01:11:13.720
So if you're looking
at mod q, everything
01:11:13.720 --> 01:11:15.348
is group homomorphism.
01:11:15.348 --> 01:11:16.140
So here we're good.
01:11:18.930 --> 01:11:21.577
Both are in this interval.
01:11:21.577 --> 01:11:23.160
The right-hand side
is in the interval
01:11:23.160 --> 01:11:26.280
because of our
assumption about short--
01:11:26.280 --> 01:11:29.220
everything living
inside a short interval.
01:11:29.220 --> 01:11:30.990
And the left-hand
side is by definition.
01:11:30.990 --> 01:11:40.730
Because the image of
phi is in that interval,
01:11:40.730 --> 01:11:43.040
especially if given that
d is not equal to 0.
01:11:47.950 --> 01:11:50.160
Is that OK?
01:11:50.160 --> 01:11:52.870
So it's not hard, but
it's a bit confusing.
01:11:52.870 --> 01:11:56.080
So think about it.
01:11:56.080 --> 01:11:56.580
All right.
01:11:56.580 --> 01:11:57.900
So we're almost done.
01:11:57.900 --> 01:12:03.540
So we're almost done
proving the Ruzsa modeling
01:12:03.540 --> 01:12:06.375
lemma in Z mod m.
01:12:06.375 --> 01:12:07.950
So let me finish off the proof.
01:12:25.400 --> 01:12:31.850
So for each non-zero d in this
iterated sumset, basically,
01:12:31.850 --> 01:12:35.970
we would like to pick a lambda
so that that map up there does
01:12:35.970 --> 01:12:38.940
what we want to do.
01:12:38.940 --> 01:12:40.650
If it doesn't do
what we want to do,
01:12:40.650 --> 01:12:45.060
then it is witnessed
by some d, like this.
01:12:45.060 --> 01:12:48.980
Those are the bad lambda.
01:12:48.980 --> 01:12:53.250
So if there exists a d,
then this lambda is bad.
01:12:53.250 --> 01:12:56.850
So for each d that
potentially witnessed
01:12:56.850 --> 01:13:09.950
some bad lambda, the
number of bad lambda, i.e.,
01:13:09.950 --> 01:13:19.086
such that phi of d
is congruent to 0 mod
01:13:19.086 --> 01:13:25.820
m, so here we're no
longer even thinking
01:13:25.820 --> 01:13:27.740
about group
homomorphisms anymore
01:13:27.740 --> 01:13:29.410
or the Freiman homomorphisms.
01:13:29.410 --> 01:13:33.460
It's just a question of, if I
give you a non-zero integer,
01:13:33.460 --> 01:13:40.540
how many lambdas are
there so that phi of d
01:13:40.540 --> 01:13:42.290
is divisible by m?
01:13:45.300 --> 01:13:50.790
Remember that we picked q large
enough so that, initially, you
01:13:50.790 --> 01:13:52.905
are sitting very much inside--
01:13:55.680 --> 01:13:57.750
everything's really
between 0 and q.
01:13:57.750 --> 01:14:05.590
So this dot lambda up
there lacks uniformity.
01:14:05.590 --> 01:14:08.220
So the number of
such bad lambdas
01:14:08.220 --> 01:14:17.790
is exactly the number of
elements in this interval that
01:14:17.790 --> 01:14:22.608
are divisible by m.
01:14:28.240 --> 01:14:30.340
So everything's more
or less a bijection
01:14:30.340 --> 01:14:31.840
if you restrict to
the right places.
01:14:35.800 --> 01:14:42.120
And the number of such elements
is, at most, q minus 1 over m.
01:14:42.120 --> 01:14:49.320
So therefore, the total
number of bad lambdas
01:14:49.320 --> 01:14:54.570
is, at most, for
each element d of sA
01:14:54.570 --> 01:14:58.740
minus sA, a non-zero element.
01:14:58.740 --> 01:15:06.520
We have, at most, q minus
1 over m bad lambdas.
01:15:06.520 --> 01:15:10.290
So the total number of bad
lambdas is strictly less than q
01:15:10.290 --> 01:15:11.860
minus 1.
01:15:11.860 --> 01:15:19.080
So there exists some
lambda such that psi,
01:15:19.080 --> 01:15:30.100
when restricted to A sub lambda,
maps Freiman s-isomorphically
01:15:30.100 --> 01:15:31.440
onto the image.
01:15:43.010 --> 01:15:45.340
Somehow, I think it's really
the same kind of proof
01:15:45.340 --> 01:15:46.840
as the one in the
finite field case,
01:15:46.840 --> 01:15:49.560
except you have this extra
wrinkle about restricting
01:15:49.560 --> 01:15:53.130
too short diameter intervals,
to short intervals.
01:15:53.130 --> 01:15:54.390
But the idea is very similar.
01:15:57.490 --> 01:15:57.990
OK.
01:15:57.990 --> 01:16:03.000
So that's the Freiman model
lemma in the integers.
01:16:03.000 --> 01:16:07.350
And let me summarize
what we know so far.
01:16:07.350 --> 01:16:09.150
And so that will
give you a sense
01:16:09.150 --> 01:16:13.210
of where we're going in the
proof of Freiman's theorem.
01:16:13.210 --> 01:16:18.160
So what we know so
far is that if you
01:16:18.160 --> 01:16:22.570
have a subset of
integers, a finite subset,
01:16:22.570 --> 01:16:27.760
such that A plus A is
size A times K at most,
01:16:27.760 --> 01:16:36.900
doubling constant at most K,
then there exists some prime N
01:16:36.900 --> 01:16:44.368
at most 2K to the 16th
times the size of A
01:16:44.368 --> 01:16:50.830
and some subset
A prime of A such
01:16:50.830 --> 01:17:04.977
that A prime is Freiman
8-isomorphic to a subset of Z
01:17:04.977 --> 01:17:05.540
mod NZ.
01:17:09.698 --> 01:17:13.970
So it follows from two
things we've seen so far.
01:17:13.970 --> 01:17:21.950
Because by the Plunnecke-Ruzsa
inequality 8A minus 8A
01:17:21.950 --> 01:17:27.146
is, at most, K to
the 16 times A.
01:17:27.146 --> 01:17:34.130
And now we can choose
a prime N between K
01:17:34.130 --> 01:17:38.950
to the 16 and 2
times K to the 16
01:17:38.950 --> 01:17:41.180
and apply the modeling lemma.
01:17:50.450 --> 01:17:52.010
So that's where we are at.
01:17:52.010 --> 01:17:56.980
So you start with a set of
integers with small doubling.
01:17:56.980 --> 01:18:02.470
Then we can conclude that by
keeping a large fraction of A,
01:18:02.470 --> 01:18:04.960
keeping--
01:18:04.960 --> 01:18:07.730
I forgot to-- so very important.
01:18:07.730 --> 01:18:10.810
So there exist some A,
which is a large fraction.
01:18:15.320 --> 01:18:19.550
Keeping a large fraction of A,
I can model this large subset
01:18:19.550 --> 01:18:25.430
of A by some subset
of a cyclic group,
01:18:25.430 --> 01:18:27.860
where the size of
the cyclic group
01:18:27.860 --> 01:18:33.550
is only a constant times
more than the size of A.
01:18:33.550 --> 01:18:41.360
So now, we are going to
work inside a cyclic group
01:18:41.360 --> 01:18:46.640
and working with a set
inside a cyclic group that's
01:18:46.640 --> 01:18:50.044
a constant size, a constant
fraction of the cyclic group.
01:18:54.000 --> 01:18:54.940
Question is why 8?
01:18:54.940 --> 01:18:56.630
So that will come up later.
01:18:56.630 --> 01:18:59.250
So basically, you need
to choose some numbers
01:18:59.250 --> 01:19:03.320
so that you want to preserve
the structure of GAPs.
01:19:03.320 --> 01:19:06.260
So that will come up later.
01:19:06.260 --> 01:19:10.010
And now we're inside
a cyclic group.
01:19:10.010 --> 01:19:13.668
And you have a constant
fraction of a cyclic group.
01:19:13.668 --> 01:19:14.960
Where have we seen this before?
01:19:20.030 --> 01:19:24.065
So when we proved Roth's
theorem, that was the setting.
01:19:24.065 --> 01:19:26.340
So in cyclic loop, you
have a constant fraction
01:19:26.340 --> 01:19:29.270
of cyclic group, and you
can do Fourier analysis.
01:19:29.270 --> 01:19:32.340
Initially, you could
not do Fourier analysis
01:19:32.340 --> 01:19:34.340
starting with Freiman's
theorem, because the set
01:19:34.340 --> 01:19:35.870
may be very, very spread out.
01:19:35.870 --> 01:19:39.410
But now, you are a large
fraction of a cyclic group.
01:19:39.410 --> 01:19:42.830
So we're going to do
Fourier analysis next time
01:19:42.830 --> 01:19:47.780
to show that such a set must
contain lots of structure,
01:19:47.780 --> 01:19:51.890
just from the fact
that it is large.
01:19:51.890 --> 01:19:54.260
So that will be the next step.
01:19:54.260 --> 01:19:54.760
Good.
01:19:54.760 --> 01:19:57.150
Happy Thanksgiving.