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, we prove that \[ R_{S_j}^*(2\to r)\lesssim_r 1 \] for every non-zero radius sphere $S_j\subseteq {\mathbb{F}}_p^4$ and every $r>23/7$, thereby breaking the Stein--Tomas exponent of $10/3$. We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture simultaneously captures the spherical restriction conjecture $L^2\to L^3$ up to the endpoint and the Erdős--Falconer distance conjecture in four dimensions. Using the same structural method, we establish the first nontrivial bounds toward this localized conjecture and derive new almost-every-pin distance estimates in ${\mathbb{F}}_p^4$.

V3: We introduce the localized spherical restriction/extension conjecture and discuss applications

Topics & keywords