Exceptional set estimates for orthogonal and radial projections in $\mathbb{R}^n$

Paige Dote, Shengwen Gan

Published 2022-08-06Version 1

We give different proofs of classic Falconer-type and Kaufman-type exceptional estimates for orthogonal projections using the high-low method. With the new techniques, we resolve Liu's conjecture on radial projections: given a Borel set $A\subset \mathbb{R}^n$, we have \[ \text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(\pi_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil. \]