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Let ω be a sequence of positive integers. Given a positive integer n, we define rₙ(ω) = |(a,b) ∈ ℕ × ℕ : a,b ∈ ω, a+b = n, 0 < a < b|. S. Sidon conjectured that there exists a sequence ω such that rₙ(ω) > 0 for all n sufficiently large and, for all ϵ > 0, $li{m}_{n\to \infty }rₙ\left(\omega \right)/n^{ϵ}=0$. P. Erdős proved this conjecture by showing the existence of a sequence ω of positive integers such that log n ≪ rₙ(ω) ≪ log n. In this paper, we prove an analogue of this conjecture in ${}_q\left[T\right]$, where ${}_q$ is a finite field of q elements....

Let ${𝔽}_q\left[t\right]$ denote the polynomial ring over ${𝔽}_q$, the finite field of $q$ elements. Suppose the characteristic of ${𝔽}_q$ is not $2$ or $3$. We prove that there exist infinitely many $N\in \mathbb{N}$ such that the set $\{f\in {𝔽}_q\left[t\right]:degf<N\}$ contains a Sidon set which is an additive basis of order $3$.