Restricted families of projections onto planes: the general case of nonvanishing geodesic curvature

Harris, Terence L. J.

doi:10.48550/arxiv.2107.14701

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2022

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“…To have a better understanding of this restricted projection problem, the first step is to study the problem in R 3 . Fässler and Orponen made a conjecture about restricted projections to lines and planes (see Conjecture 1.6 in [4]), and there has been much related research (see for example [4], [2], [14], [13], [15], [19], [20], [21], [11], [12]). For more of an introduction to this problem, we refer to [11].…”

Section: Introductionmentioning

confidence: 99%

On restricted projections to planes in $\mathbb{R}^3$

Gan¹,

Guo²,

Guth³

et al. 2022

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Let γ : [0, 1] → S 2 be a non-degenerate curve in R 3 , 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 : R 3 → V θ be the orthogonal projections. We prove that if A ⊂ R 3 is a Borel set, then for a.e. θ ∈ [0, 1] we have dim(π θ (A)) = min{2, dimA}.More generally, we prove an exceptional set estimate. For A ⊂ R 3 and 0 ≤ s ≤ 2, define Es(A) := {θ ∈ [0, 1] : dim(π θ (A)) < s}. We have dim(Es(A)) ≤ 1 + s − dim(A). We also prove that if dim(A) > 2, then for a.e. θ ∈ [0, 1] we have H 2 (π θ (A)) > 0.2020 Mathematics Subject Classification. 42B15, 42B20.

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