P-adic Curved Kakeya Sets, Projection Theorem and Covering Numbers
Abstract
Let n,d≥1, and suppose that E⊂ ℚpn contains a degree-d polynomial image of ℤp in every leading direction of a nonempty set V⊂ℙn−1(ℚp). We prove the sharp bound dimH E≥dimH V+1. For joint block-valued polynomial evaluations, the total incidence dimension is at least one plus the supremum of the image dimensions. For analytic coefficient sets, Haar-almost every evaluation attains this supremum, and we bound the Hausdorff dimension of the parameters where the image dimension drops. For bounded coefficient sets and arbitrary fixed anisotropic block scales, we prove a quantitative covering comparison with a common Haar-null exceptional set. Outside this set, each evaluation's covering number dominates the maximum over any compact reference ball, up to any positive power loss at all sufficiently small scales. The proofs use the polynomial-to-line lifting map due to Nadjimzadah, together with Dhar's prime-power Kakeya set estimate and a tube maximal inequality derived from it.
Related
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BibTeX
@misc{lou2026padic,
title = {{P-adic Curved Kakeya Sets, Projection Theorem and Covering Numbers}},
author = {Yi Lou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22485},
eprint = {2609.22485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22485},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - P-adic Curved Kakeya Sets, Projection Theorem and Covering Numbers
AU - Yi Lou
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.22485
UR - https://arxiv.org/abs/2609.22485
AB - Let n,d≥1, and suppose that E⊂ ℚ_p^n contains a degree-d polynomial image of ℤ_p in every leading direction of a nonempty set V⊂ℙ^n−1(ℚ_p). We prove the sharp bound dim_H E≥dim_H V+1. For joint block-valued polynomial evaluations, the total incidence dimension is at least one plus the supremum of the image dimensions. For analytic coefficient sets, Haar-almost every evaluation attains this supremum, and we bound the Hausdorff dimension of the parameters where the image dimension drops. For bounded coefficient sets and arbitrary fixed anisotropic block scales, we prove a quantitative covering comparison with a common Haar-null exceptional set. Outside this set, each evaluation's covering number dominates the maximum over any compact reference ball, up to any positive power loss at all sufficiently small scales. The proofs use the polynomial-to-line lifting map due to Nadjimzadah, together with Dhar's prime-power Kakeya set estimate and a tube maximal inequality derived from it.
ER -