Number Theory Seminar

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.