On a problem of Sidon for polynomials over finite fields

Abstract

Let $\omega$ be a sequence of positive integers. Given a positive integer $n$, we define $$ r_n(\omega) = | \{ (a,b)\in \mathbb{N}\times \mathbb{N}\colon a,b \in \omega, a+b = n, 0 <a<b \}|. $$ S. Sidon conjectured that there exists a sequence $\omega$ such that $r_n(\omega) > 0$ for all $n$ sufficiently large and, for all $\epsilon > 0$, $$ \lim_{n \rightarrow \infty} \frac{r_n(\omega)}{n^{\epsilon}} = 0. $$ P. 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 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 $$ r_h(\omega) = |\{(f,g) \in \mathbb{F}_q[T]\times \mathbb{F}_q[T] : f,g\in \omega, f+g =h, \text{deg } f, \text{deg } g \leq \text{deg } h, f\ne g\}|. $$ We show that there exists a sequence $\omega$ of polynomials in $\mathbb{F}_q [T]$ such that $$ \text{deg } h \ll r_h(\omega) \ll \text{deg } h $$ for $\text{deg } h$ sufficiently large.