On a problem of Sidon for polynomials over finite fields
Volume 174 / 2016
Abstract
Let $\omega$ be a sequence of positive integers. Given a positive integer $n$, we define \[ r_n(\omega) = | \{ (a,b)\in \mathbb{N}\times \mathbb{N}: a,b \in \omega,\, a+b = n,\, 0 \lt a \lt b \}|. \] S. Sidon conjectured that there exists a sequence $\omega$ such that $r_n(\omega) \gt 0$ for all $n$ sufficiently large and, for all $\epsilon \gt 0$, \[ \lim_{n \rightarrow \infty} \frac{r_n(\omega)}{n^{\epsilon}} = 0. \] P. Erdős 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 paper, 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 \[ r_h(\omega) = |\{(f,g) \in \mathbb{F}_q[T]\times \mathbb{F}_q[T] : f,g\in \omega,\, f+g =h, \deg f, \deg g \leq \deg h,\, f\ne g\}|. \] 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 $\deg h$ tending to infinity.
