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

Abstract

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.

Length of sets under restricted families of projections onto lines

“…followed by Lemma 5 from [2] with k = j, the contribution from (2.5) is 2 −J µ(R 3 ) ∼ R −ǫ/1000 µ(R 3 ).…”

“…The proof of Theorem 1.2 is similar to the proof of Theorem 8 in [2], which solved the analogous problem of projections onto planes. It uses a variant of the "good-bad" decomposition of a measure, which originated in [3].…”

An exceptional set estimate for restricted projections to lines in $\mathbb{R}^3$

“…Next, we state a useful lemma. It is proved in [3], but we still provide the full details here. The lemma roughly says that given a set X of Hausdorff dimension less than s, then we can find a covering of X by squares of dyadic lengths which satisfy a certain s-dimensional condition.…”

A note on Kakeya sets of horizontal and SL(2)$SL(2)$ lines

“…The purpose of this note is to study the Hausdorff dimension of unions of 𝑆𝐿(2) lines in ℝ 3 . Here is the definition of 𝑆𝐿(2) lines, following [10].…”

“…Definition 1.1 ( 𝑆𝐿 (2) ). The family  𝑆𝐿(2) consists of the following lines 𝐿 ⊂ ℝ 3 . Either 𝐿 is a line contained in the 𝑥𝑦-plane, and 0 ∈ 𝐿, or then 𝐿 ∶= 𝐿 𝛼,𝛽,𝛾,𝛿 ∶= (𝛼, 𝛽, 0) + span(𝛾, 𝛿, 1),…”