On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Abstract

We introduce a method based on horizontal slicing and a plane-then-line stopping-time decomposition for the prime field spherical restriction problem in four dimensions. The method is designed to overcome the Kloosterman obstruction in the spherical Bochner--Riesz kernel by decomposing each critical horizontal slice into rich-plane, rich-line-and-poor-plane, and poor-line-and-poor-plane components, which are then treated by distinct affine-geometric mechanisms. As a quantitative consequence of this structural method, we prove that \[ R_{S_j}^*(2\to r)\lesssim 1 \] for every nonzero sphere $S_j\subset\mathbb{F}^4$ and every $r>23/7$. As an application, we obtain the first improvement over the twenty-year-old $(d+1)/2$ threshold in the four-dimensional prime field Erdős-Falconer distance problem.