A Prime Analogue of Roth’s Theorem in Function Fields

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 = (r1, r2, r3) of \(\mathbb{F}_{q}\) satisfying r1 + r2 + r3 = 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 \(D(\mathcal{P}_{R}) \ll _{q}\vert \mathcal{P}_{R}\vert /(\log \log \log \log \vert \mathcal{P}_{R}\vert )\).