An improved non-linear Roth-type theorem in finite fields

Let $F$ be a finite field of odd characteristic. We prove that any set $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains a nontrivial quadratic progression $(x, x+y, x+y^2), y\neq 0.$ For prime fields, this improves the previous best-known exponent of $7/8$, due to Kavrut and Wu. Unlike some of the previous papers, which rely on Katz's deep multivariate exponential-sum estimates, our argument uses only one-variable Weil-type estimates. We also construct, over certain non-prime finite fields, progression-free sets of size $c|F|^{2/3}$. A key idea in the proof was suggested to the author by ChatGPT 5.5.