A Prime Analogue of Roth’s Theorem in Function Fields

Abstract

Let $\mathbb{F}_{q}[t]$ denote the polynomial ring over the finite field $\mathbb{F}_{q}$, and let $\mathcal{P}_{R}$ denote the subset of $\mathbb{F}_{q}[t]$ containing all monic irreducible polynomials of degree R. For non-zero elements r = (r ₁, r ₂, r ₃) of $\mathbb{F}_{q}$ satisfying r ₁ + r ₂ + r ₃ = 0, let $D(\mathcal{P}_{R}) = D_{\mathbf{r}}(\mathcal{P}_{R})$ denote the maximal cardinality of a set $A_{R} \subseteq \mathcal{P}_{R}$ which contains no non-trivial solution of $r_{1}x_{1} + r_{2}x_{2} + r_{3}x_{3} = 0$ with x _i  ∈ A _R  (1 ≤ i ≤ 3). By applying the polynomial Hardy-Littlewood circle method, we prove that $D(\mathcal{P}_{R}) \ll _{q}\vert \mathcal{P}_{R}\vert /(\log \log \log \log \vert \mathcal{P}_{R}\vert )$.