On exceptional sets of radial projections

Abstract

We prove two new exceptional set estimates for radial projections in the plane.The finite field counterparts of both results above were recently proven by Lund, Thang, and Huong Thu. Our results resolve the planar cases of conjectures of Lund-Thang-Huong Thu, and Liu. CONTENTS1. Introduction 1 1.1. Sharper results for sets avoiding lines? 3 1.2. Paper outline 4 1.3. Notation 4 Acknowledgements 4 2. Reduction to a δ-discretised problem 4 2.1. Proof of Proposition 2.3 6 3. Proof of Theorem 2.5 12 4. Proof of Theorem 1.7 17 References 24

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

“…where the last inequality is by the s-dimensional condition of T V . Plugging this back into (12), we obtain…”

“…Recently, Orponen and Shmerkin [12] proved the n = 2 case for both Theorem 1 and Theorem 2. Their proof of Theorem 1 (when n = 2) is based on a Furstenbergtype estimate due to Fu and Ren [3].…”

“…Here, Π is an (n − 1)-plane. Such comparison between orthogonal projection and radial projection was discussed in Orponen and Shmerkin's paper (see [12] (1.4)).…”

“…Theorem 1 is a result of Proposition 3 and Proposition 4. Finally, the proof of Theorem 2 is a combination of Theorem 1 and a trick of Orponen and Shmerkin [12].…”

“…i (see also [12] Definition 2.2). There is a natural correspondence between such planks and the δ-balls in A(n, n− m).…”