A Continuum Beck-type Theorem for Hyperplanes

We prove a sharp continuum Beck-type theorem for hyperplanes. Our work is inspired by foundational work of Beck on the discrete problem, as well as refinements due to Do and Lund. The inductive proof uses recent breakthrough results in projection theory by Orponen–Shmerkin–Wang and Ren, who proved continuum Beck-type theorems for lines in ${\mathbb{R}}^2$ and ${\mathbb{R}}^n$.