Number Theory Seminar

Thursday, October 6, 2016 1:30 pm - 1:30 pm EDT (GMT -04:00)

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
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
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\}|.
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