On a problem of Sidon for polynomials over finite fields

TL;DR

This paper extends a classical problem about the distribution of sums in integer sequences to polynomials over finite fields, proving the existence of sequences with controlled sum representations.

Contribution

It establishes an analogue of Sidon's conjecture for polynomials over finite fields, showing sequences with sum representation counts between linear bounds in degree.

Findings

Existence of polynomial sequences with sum counts between degree bounds

Analogues of Sidon's conjecture in finite field polynomial setting

Bounds on the number of representations for large degrees

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.