A bilinear approach to the finite field restriction problem, II

Abstract

Let $P_3$ denote the three-dimensional paraboloid over a finite field of prime order in which $-1$ is not a square. We prove that the Fourier extension operator associated with $P_3$ maps $L^2$ to $L^r$ for $r>\frac{176}{51}=3.45098\ldots$. The argument combines the author's bilinear approach to the problem with point-line incidence estimates. We also prove that the extension operator associated with the paraboloid $P_6$ in six dimensions maps $L^2$ to $L^{8/3}$. This was previously known up to but not including the endpoint, and is the sharp $L^2$ estimate in six dimensions. Finally we observe that the endpoint restriction conjecture for $P_3$ in finite fields implies that the integer lattice points on the $3$-d Euclidean paraboloid are a $Λ(3)$ set.