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 $\mathrm{\Pi }$ be a $\delta$-separated $(\delta ,t,C_{\mathrm{\Pi }})$-set of hyperplanes intersecting $B(0,1)$ in ${\mathbb{R}}^d$. Define $$I_{C\delta }(P,\mathrm{\Pi })=\mathrm{\#}\{(p,\pi )\in P\times \mathrm{\Pi }\text{\hspace{0.17em}⁣}:p\in \pi (C\delta )\}\mathrm{.}$$ Suppose that $s,t\ge \frac{d+1}{2}$, then we have $I_{C\delta }(P,\mathrm{\Pi })\lesssim \delta \mathrm{\mid }P\mathrm{\mid }\mathrm{\mid }\mathrm{\Pi }\mathrm{\mid }$. 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.