New Restriction Estimates for the 3-d Paraboloid over Finite Fields

TL;DR

This paper advances the restriction problem for the 3-dimensional paraboloid over finite fields by leveraging a novel incidence theorem derived from sum-product estimates, establishing explicit exponent relationships.

Contribution

Introduces a new finite field incidence theorem and links it explicitly to restriction estimates for the paraboloid over prime fields.

Findings

Extended the range of exponents for restriction estimates

Established a direct relationship between incidence bounds and restriction exponents

Demonstrated improvements specifically over prime order finite fields

Abstract

We improve the range of exponents for the restriction problem for the 3-d paraboloid over finite fields. The key new ingredient is a variant of the Bourgain-Katz-Tao finite field incidence theorem derived from sum-product estimates. In prime order fields, we give an explicit relationship between the exponent in this incidence theorem and restriction estimates for the paraboloid.