A Discretized Point-Hyperplane Incidence Bound in Rd
Journal
The Journal of Geometric Analysis
Journal Volume
35
Journal Issue
11
ISSN
1050-6926
1559-002X
Date Issued
2025-09-04
Author(s)
Abstract
Let P be a δ-separated (δ,s,CP)-set of points in B(0,1)⊂Rd and Π be a δ-separated (δ,t,CΠ)-set of hyperplanes intersecting B(0, 1) in Rd. Define (Formula presented.) where π(Cδ) denotes the Cδ neighborhood of the hyperplane π. Suppose that s,t≥d+12, then we have ICδ(P,Π)≲δ|P
Π|. 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.
Publisher
Springer Science and Business Media LLC
Type
journal article