A bilinear approach to the finite field restriction problem

TL;DR

This paper establishes new bounds for the Fourier extension operator over a finite field paraboloid using a bilinear approach, avoiding incidence estimates and leveraging geometric decompositions.

Contribution

It introduces a novel bilinear method to analyze the finite field restriction problem, providing bounds without relying on incidence geometry.

Findings

Fourier extension operator maps L^2 to L^r for r > 32/9

Geometric decomposition of point sets into rectangles and trapezoids

New bounds achieved without incidence estimates

Abstract

Let $P$ denote the $3$-dimensional paraboloid over a finite field of odd characteristic in which $-1$ is not a square. We show that the Fourier extension operator associated with $P$ maps $L^2$ to $L^{r}$ for $r > \frac{32}{9} \approx 3.555$. In contrast with much of the recent progress on this problem, our argument does not use state-of-the-art incidence estimates but rather proceeds by obtaining estimates on a related bilinear operator. These estimates are based on a geometric result that, roughly speaking, states that a set of points in the finite plane $F^2$ can be decomposed as a union of sets each of which either contains a controlled number of rectangles or a controlled number of trapezoids.