On the restriction problem for discrete paraboloid in lower dimension

TL;DR

This paper establishes optimal Fourier extension estimates for the paraboloid in four dimensions over finite fields using geometric incidence methods, and improves bounds in three dimensions under certain conditions.

Contribution

It introduces new Fourier extension bounds for paraboloids in finite fields, utilizing geometric incidence estimates in positive characteristic, with improvements in three-dimensional cases.

Findings

Optimal $L^2 o L^3$ extension estimate in 4D over prime fields.

Improved $L^2 o L^{32/9}$ estimate in 3D when -1 is not a square.

Application of geometric incidence techniques in positive characteristic settings.

Abstract

We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when $-1$ is not a square, we prove an $L^2 \to L^{\frac{32}{9} }$ extension estimate, improving the previously known exponent $\frac{68}{19}.$