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

Abstract

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 \[I_{Cδ}(P, Π)=\#\{(p, π)\in P\times Π\colon p\in π(Cδ)\}.\] 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.

14 pages