On restricted projections to planes in ℝ³

2022/07/28 by Gan, Shengwen, Guo, Shaoming, Guth, Larry +3
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2207.13844

Abstract

Let γ:[0,1]→ \mathbbS² be a non-degenerate curve in ℝ³, that is to say, det(γ(θ),γ'(θ),γ"(θ))≠ 0. For each θ∈[0,1], let V_θ=γ(θ)^⊥ and let π_θ:ℝ³→ V_θ be the orthogonal projections. We prove that if A⊂ ℝ³ is a Borel set, then for a.e. θ∈ [0,1] we have dim(π_θ(A))=min\2,dim A\. More generally, we prove an exceptional set estimate. For A⊂ℝ³ and 0≤ s≤ 2, define E_s(A):=\θ∈[0,1]: dim(π_θ(A))2, then for a.e. θ∈[0,1] we have H²(π_θ(A))>0.

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