Sharp spherical extension theorem in 𝔽_q^2m and applications

Abstract

Let q be an odd prime power and m≥ 1. Let Q be a nondegenerate quadratic form on 𝔽_q^2m. For every sphere S_j={Q=j} with j∈ 𝔽_q^×, we prove the sharp extension estimate R_{S_j}^*(2→ r)≲_m1 for r≥ 2(m+1)/m, uniformly in q, Q, and j. As an application, we show that if E⊆𝔽_q^2m satisfies |E|/q^m+1/3→∞, then almost every pin y∈ E determines $(1-o(1))q$ values of Q(x−y). The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere--cone estimate, and an orthogonal decomposition.

Sharp spherical L^p→ L⁴ extension estimates over finite fields in odd dimensions
Thang Pham, Boqing Xue
arXiv21 Oct 2026PreprintarXiv licenceFull textAlgebra & number theory Analysis & PDEsarXiv
A Sharp Surface-Area Extension of Vaaler's Theorem
Michał Zwierzyński
arXiv1 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEs Geometry & topologyarXiv
On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
Thang Pham, Boqing Xue
arXiv29 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Analysis & PDEsarXiv
A Second-Order Extension of Hájek's Convolution Theorem with Statistical Applications
Junichi Hirukawa, Masanobu Taniguchi, Marc Hallin
arXiv26 Sep 2026PreprintarXiv licenceFull textStatisticsarXiv
A CRANDALL-PAZY EXTENSION OF THE SLOPE ATTOUCH THEOREM
Julien Bernis
HAL12 Sep 2026PreprintPublisher termsFull textOptimisation & controlHAL

Cite

BibTeX RIS

BibTeX

@misc{pham2026sharp,
  title = {{Sharp spherical extension theorem in $\mathbb F_q^2m$ and applications}},
  author = {Thang Pham and Boqing Xue},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39876},
  eprint = {2609.39876},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39876},
}

RIS

TY  - UNPB
TI  - Sharp spherical extension theorem in 𝔽_q^2m and applications
AU  - Thang Pham
AU  - Boqing Xue
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39876
UR  - https://arxiv.org/abs/2609.39876
AB  - Let q be an odd prime power and m≥ 1. Let Q be a nondegenerate quadratic form on 𝔽_q^2m. For every sphere S_j={Q=j} with j∈ 𝔽_q^×, we prove the sharp extension estimate R_S_j^*(2→ r)≲_m1 for r≥ 2(m+1)/m, uniformly in q, Q, and j. As an application, we show that if E⊆𝔽_q^2m satisfies |E|/q^m+1/3→∞, then almost every pin y∈ E determines $(1-o(1))q$ values of Q(x−y). The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere--cone estimate, and an orthogonal decomposition.
ER  -

Abstract

Related

Cite