Finite field restriction estimates based on Kakeya maximal operator estimates

Abstract

In the finite field setting, we show that the restriction conjecture associated to any one of a large family of d = 2n + 1 dimensional quadratic surfaces implies the n + 1 dimensional Kakeya conjecture (Dvir’s theorem). This includes the case of the paraboloid over finite fields in which −1 is a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein-Tomas exponent in this setting.