A bilinear approach to the finite field restriction problem

TL;DR

This work advances the finite-field restriction problem for the 3D paraboloid by establishing an $L^2\to L^r$ bound for all $r>\frac{24}{7}$ in odd-characteristic fields with $-1$ non-square. It introduces a bilinear framework that reduces the problem to estimating a bilinear operator tied to trapezoid and rectangle configurations in the plane, bypassing deep incidence theorems. A new geometric decomposition of planar sets into $k$-regular components and corresponding trapezoid/rectangle bounds yields improved estimates, culminating in a dual form $||\hat g||_{L^{2}(P,d\sigma)} \lesssim ||g||_{L^{24/17-\varepsilon}(F^3)}$, which implies the claimed $L^r$ result. The approach highlights a near-optimal interplay between combinatorial geometry and harmonic analysis in the finite-field setting and may inform extensions to higher dimensions or alternative norm regimes.

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 $r > \frac{24}{7} \approx3.428$. Previously this was known (in the case of prime order fields) for $r > \frac{188}{53} \approx 3.547$. 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.

Theorems & Definitions (23)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• Lemma 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• Proposition 4.1