On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
Year: 2026
Type: preprint
Abstract: Let $p$ be an odd prime. We prove the sharp extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every sphere of nonzero radius $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq3$, uniformly in $p$ and $j$. The proof combines arithmetic Hecke operator bounds with a refined fourth moment and an orthogonal decomposition. We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence on the size of the physical support. This conjecture remains open and would yield distance estimates for almost every pin at the conjectured Erdős--Falconer exponent in four dimensions, with an arbitrarily small power loss in the size hypothesis. Unconditionally, our localized estimates imply that, for every $\varepsilon>0$, sets of size at least $p^{7/3+\varepsilon}$ determine $(1+o_\varepsilon(1))p$ distances from almost every pin, together with asymmetric two-set versions.
Source: arXiv (Cornell University)
Authors: Pham, Thang, Xue, Boqing
Institutions: Xidian University, ShanghaiTech University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7163336205