A bilinear approach to the finite field restriction problem

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.

12 pages, no figures, v4: Erratum added. The published version of this manuscript claimed the stronger range $r>\frac{24}{7}$. The argument up to the final exponent calculation remains valid, but the claimed range was misstated due to a computation error. This arXiv version has been updated to reflect the corrected final calculation and resulting range