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 $\mathrm{\mid }A\mathrm{\mid }\ge C\mathrm{\mid }F{\mathrm{\mid }}^{5\mathrm{/}6}$ contains a nontrivial quadratic progression $(x,x+y,x+y^2),y\mathrm{\ne }0.$ For prime fields, this improves the previous best-known exponent of $7\mathrm{/}8$, 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 $c\mathrm{\mid }F{\mathrm{\mid }}^{2\mathrm{/}3}$. A key idea in the proof was suggested to the author by ChatGPT 5.5.