On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
Abstract
Let p be an odd prime. We prove the sharp extension estimate RSj*(2→ r)≲r 1 for every sphere of nonzero radius Sj⊆𝔽p4 and every r≥3, 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 ε>0, sets of size at least p7/3+ε determine (1+oε(1))p distances from almost every pin, together with asymmetric two-set versions.
Related
- Endpoint eigenfunction restriction estimates in codimension two
- Endpoint power weight inequalities for spherical maximal functions
- Besicovitch Compression and a Quantitative Failure of Endpoint Fourier Restriction
- Endpoint weighted Fourier estimate with vanishing spherical trace
- Strong-type endpoint obstructions for bilinear spherical maximal operators
BibTeX
@misc{pham2026finite,
title = {{On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications}},
author = {Thang Pham and Boqing Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.03627},
eprint = {2606.03627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.03627},
}
RIS
TY - UNPB
TI - On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
AU - Thang Pham
AU - Boqing Xue
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2606.03627
UR - https://arxiv.org/abs/2606.03627
AB - Let p be an odd prime. We prove the sharp extension estimate R_S_j^*(2→ r)≲_r 1 for every sphere of nonzero radius S_j⊆𝔽_p^4 and every r≥3, 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 ε>0, sets of size at least p^7/3+ε determine (1+o_ε(1))p distances from almost every pin, together with asymmetric two-set versions.
ER -