Wentang Kuo, Pure Mathematics, University of Waterloo
"On a problem of Sidon"
Let
\(\omega\)
be
a
sequence
of
positive
integers.
Given
a
positive
integer
\(n\),
we
define
\[
r_n(\omega)
=

\{
(a,b)\in
\mathbb{N}\times
\mathbb{N}\colon
a,b
\in
\omega,
a+b
=
n,
0
<a<b
\}.
\]
Sidon
conjectured
that
there
exists
a
sequence
\(\omega\)
such
that
\(r_n(\omega)
>
0\)
for
all
\(n\)
sufficiently
large
and,
for
all
\(\epsilon
>
0\),
\[
\lim_{n
\rightarrow
\infty}
\frac{r_n(\omega)}{n^{\epsilon}}
=
0.
\]
Erd\H{o}s
proved
this
conjecture
by
showing
the
existence
of
a
sequence
\(\omega\)
of
positive
integers
such
that
\[
\log
n
\ll
r_n(\omega)
\ll
\log
n.
\]
In
this
talk,
we
prove
an
analogue
of
this
conjecture
in
$\mathbb{F}_q[T]$,
where
\(\mathbb{F}_q\)
is
a
finite
field
of
$q$
elements.
More
precisely,
let
\(\omega\)
be
a
sequence
in
\(\mathbb{F}_q[T]\).
Given
a
polynomial
\(h\in\mathbb{F}_q[T]\),
we
define
\[
\begin{split}
r_h(\omega)
&
=
\{(f,g)
\in
\mathbb{F}_q[T]\times
\mathbb{F}_q[T]
:
f,g\in
\omega,
f+g
=h,
\\
&
\deg
f,
\deg
g
\leq
\deg
h,
f\ne
g\}.
\end{split}
\]
We
show
that
there
exists
a
sequence
\(\omega\)
of
polynomials
in
\(\mathbb{F}_q
[T]\)
such
that
\[
\deg
h
\ll
r_h(\omega)
\ll
\deg
h
\]
for
\(\deg
h\)
sufficiently
large.
This
is
a
joint
work
with
Shuntaro
Yamagishi.