Sidon basis in polynomial rings over finite fields

Abstract

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. Suppose the characteristic of $\mathbb{F}_q$ is not $2$ or $3$. In this paper, we prove an $\mathbb{F}_q[t]$-analogue of results related to the conjecture of Erdős on the existence of infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We prove that there exists a $B_2[2]$ sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is an asymptotic basis of order $3$. We also prove that for any $\varepsilon> 0$, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is a Sidon basis of order $3 + \varepsilon$. In other words, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$ such that any $n \in \mathbb{F}_q[t]$ of sufficiently large degree can be expressed as a sum of four elements of the sequence, where one of them has a degree less than or equal to $\varepsilon \text{deg } n.$