Chao Lin, Department of Pure Mathematics, University of Waterloo
"Freiman's Theorem - Part I"
For
a
set
of
numbers
$A$,
let
the
sum-set
$A+A$
denote
$\{a_1+a_2:
a_1,
a_2
\in
A
\}$.
Freiman's
theorem
proves
the
remarkable
notion
that
if
a
finite
subset
of
integers
$A$
has
a
relatively
small
sum-set
$A+A$,
then
$A$
essentially
resembles
an
arithmetic
progression.
More
precisely,
if
$|A+A|
\leq
C|A|$
then
there
exists
constants
$d,
S$
depending
only
on
$C$
such
that
$A$
is
contained
in
a
generalized
arithmetic
progression
of
dimension
$d$
and
size
$\leq
S|A|$.
This
is
the
first
of
two
talks
presenting
the
proof
of
Freiman's
theorem,
which
uses
ideas
ranging
from
graph
theory,
discrete
Fourier
analysis,
and
Minkowski's
geometry
of
numbers.
For
a
set
of
numbers
$A$,
let
the
sum-set
$A+A$
denote
$\{a_1+a_2:
a_1,
a_2
\in
A
\}$.
Freiman's
theorem
proves
the
remarkable
notion
that
if
a
finite
subset
of
integers
$A$
has
a
relatively
small
sum-set
$A+A$,
then
$A$
essentially
resembles
an
arithmetic
progression.
More
precisely,
if
$|A+A|
\leq
C|A|$
then
there
exists
constants
$d,
S$
depending
only
on
$C$
such
that
$A$
is
contained
in
a
generalized
arithmetic
progression
of
dimension
$d$
and
size
$\leq
S|A|$.
This
is
the
first
of
two
talks
presenting
the
proof
of
Freiman's
theorem,
which
uses
ideas
ranging
from
graph
theory,
discrete
Fourier
analysis,
and
Minkowski's
geometry
of
numbers.