PreprintCC BY 4.0arXiv
A bilinear approach to the finite field restriction problem, II
Abstract
Let P3 denote the three-dimensional paraboloid over a finite field of prime order in which −1 is not a square. We prove that the Fourier extension operator associated with P3 maps L2 to Lr for r>176/51=3.45098…. The argument combines the author's bilinear approach to the problem with point-line incidence estimates. We also prove that the extension operator associated with the paraboloid P6 in six dimensions maps L2 to L8/3. This was previously known up to but not including the endpoint, and is the sharp L2 estimate in six dimensions. Finally we observe that the endpoint restriction conjecture for P3 in finite fields implies that the integer lattice points on the $3$-d Euclidean paraboloid are a Λ(3) set.
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BibTeX
@misc{lewko2026bilinear,
title = {{A bilinear approach to the finite field restriction problem, II}},
author = {Mark Lewko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.22882},
eprint = {2606.22882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.22882},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A bilinear approach to the finite field restriction problem, II
AU - Mark Lewko
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2606.22882
UR - https://arxiv.org/abs/2606.22882
AB - Let P_3 denote the three-dimensional paraboloid over a finite field of prime order in which −1 is not a square. We prove that the Fourier extension operator associated with P_3 maps L^2 to L^r for r>176/51=3.45098…. The argument combines the author's bilinear approach to the problem with point-line incidence estimates. We also prove that the extension operator associated with the paraboloid P_6 in six dimensions maps L^2 to L^8/3. This was previously known up to but not including the endpoint, and is the sharp L^2 estimate in six dimensions. Finally we observe that the endpoint restriction conjecture for P_3 in finite fields implies that the integer lattice points on the $3$-d Euclidean paraboloid are a Λ(3) set.
ER -