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 $$r_n(\omega )=|\{(a,b)\in \mathbb{N}\times \mathbb{N}:a,b\in \omega ,a+b=n,0<a<b\}|.$$ S. Sidon conjectured that there exists a sequence $\omega$ such that $r_n(\omega )>0$ for all $n$ sufficiently large and, for all $\epsilon >0$, $$\underset{n\to \infty }{\lim }\frac{r_n(\omega )}{n^{\epsilon }}=0.$$ 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 $$r_h(\omega )=|\{(f,g)\in {\mathbb{F}}_q[T]\times {\mathbb{F}}_q[T]:f,g\in \omega ,f+g=h,\text{deg }f,\text{deg }g\le \text{deg }h,f\ne g\}|.$$ 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.