Finite field restriction estimates for the paraboloid in high even dimensions

TL;DR

This paper establishes optimal finite field restriction estimates for the paraboloid in high even dimensions, improving known bounds and employing new additive energy techniques.

Contribution

It provides the first optimal $L^2$ restriction estimates for the paraboloid in even dimensions $d ext{ } extgreater=8$, and improves bounds in lower dimensions using additive energy methods.

Findings

Optimal $L^2 o L^r$ bounds for $d ext{ } extgreater=8$

Improved bounds for $d=6$ and $d=4$

Introduction of enhanced additive energy estimates

Abstract

We prove that the finite field Fourier extension operator for the paraboloid is bounded from $L^2\to L^r$ for $r\geq \frac{2d+4}{d}$ in even dimensions $d\ge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimal range $r> \frac{2d+4}{d}=8/3$, apart from the endpoint. For $d=4$ we improve the prior range of $r>16/5=3.2$ to $r\geq 28/9=3.111\ldots$, compared to the conjectured range of $r\geq3$. The key new ingredient is improved additive energy estimates for subsets of the paraboloid.