A discretized point-hyperplane incidence bound in \(\mathbb{R}^d\)

Let \(P\) be a \(\delta\)-separated \((\delta, s, C_P)\)-set of points in \(B(0, 1)\subset \mathbb{R}^d\) and \(\Pi\) be a \(\delta\)-separated \((\delta, t, C_\Pi)\)-set of hyperplanes intersecting \(B(0, 1)\) in \(\mathbb{R}^d\). Define \[I_{C\delta}(P, \Pi)=\#\{(p, \pi)\in P\times \Pi\colon p\in...

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Bibliographic Details
Published in:arXiv.org 2023-04
Main Authors: Pham, Thang, Chun-Yen, Shen, Nguyen Pham Minh Tri
Format: Article
Language:English
Subjects:
Hyperplanes
Radon transformation
Online Access:Get full text
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Summary:Let \(P\) be a \(\delta\)-separated \((\delta, s, C_P)\)-set of points in \(B(0, 1)\subset \mathbb{R}^d\) and \(\Pi\) be a \(\delta\)-separated \((\delta, t, C_\Pi)\)-set of hyperplanes intersecting \(B(0, 1)\) in \(\mathbb{R}^d\). Define \[I_{C\delta}(P, \Pi)=\#\{(p, \pi)\in P\times \Pi\colon p\in \pi(C\delta)\}.\] Suppose that \(s, t\ge \frac{d+1}{2}\), then we have \(I_{C\delta}(P, \Pi)\lesssim \delta |P||\Pi|\). The main ingredient in our argument is a measure theoretic result due to Eswarathansan, Iosevich, and Taylor (2011) which was proved by using Sobolev bounds for generalized Radon transforms. Our result is essentially sharp, a construction will be provided and discussed in the last section.
ISSN:2331-8422