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

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 $$R_{S_j}^{\ast }(2\to r)\lesssim 1$$ for every nonzero sphere $S_j\subset {\mathbb{F}}^4$ and every $r>23\mathrm{/}7$. As an application, we obtain the first improvement over the twenty-year-old $(d+1)\mathrm{/}2$ threshold in the four-dimensional prime field Erdős-Falconer distance problem. A series of applications is obtained, including improved Erdős–Falconer and pinned-distance estimates in ${\mathbb{F}}_p^4$, almost-every-pin results for determinant differences in $M_2({\mathbb{F}}_p)$ and dot products on the paraboloid $P_5$, and results on sparse and stable signal recovery from spherical frequency erasures.