On a problem of Sidon for polynomials over finite fields

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

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