2022/05/27 by Tuomas Orponen, Pablo Shmerkin, Orponen, Tuomas +1
Mathematics · #27A80 #28A78 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2205.13890
openalex publication_date 2022/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove two new exceptional set estimates for radial projections in the plane. If K ⊂ ℝ2 is a Borel set with dimH K > 1, then dimH \x ∈ ℝ2 ∖ K : dimH πx(K) ≤ σ\ ≤ max\1 + σ- dimH K,0\, σ∈ [0,1). If K ⊂ ℝ2 is a Borel set with dimH K ≤ 1, then dimH \x ∈ ℝ2 ∖ K : dimH πx(K) lt; dimH K\ ≤ 1. 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.