On the Finite Field Cone Restriction Conjecture in Four Dimensions and Applications in Incidence Geometry

The first purpose of this paper is to solve completely the finite field cone restriction conjecture in four dimensions with $-1$ non-square. The second is to introduce a new approach to study incidence problems via restriction theory. More precisely, using the cone restriction estimates, we will prove sharp point-sphere incidence bounds associated with complex-valued functions for sphere sets of small size. Our incidence bounds with a specific function improve significantly a result given by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev.