Length of sets under restricted families of projections onto lines

Harris, Terence L. J.

doi:10.48550/arxiv.2208.06896

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“…which consists of directions to which the projection of A has zero measure. We notice that recently Harris [3]…”

Section: Remarkmentioning

confidence: 98%

An Exceptional Set Estimate for Restricted Projections to Lines in $$\mathbb R^3$$

Gan,

Guth,

Maldague

2023

J Geom Anal

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Let $$\gamma :[0,1]\rightarrow \mathbb S^{2}$$ γ : [ 0 , 1 ] → S 2 be a non-degenerate curve in $$\mathbb R^3$$ R 3 , that is to say, $$\det \big (\gamma (\theta ),\gamma '(\theta ),\gamma ''(\theta )\big )\ne 0$$ det ( γ ( θ ) , γ ′ ( θ ) , γ ′ ′ ( θ ) ) ≠ 0 . For each $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] , let $$l_\theta =\text {span}(\gamma (\theta ))$$ l θ = span ( γ ( θ ) ) and $$\rho _\theta :\mathbb R^3\rightarrow l_\theta $$ ρ θ : R 3 → l θ be the orthogonal projections. We prove an exceptional set estimate. For any Borel set $$A\subset \mathbb R^3$$ A ⊂ R 3 and $$0\le s\le 1$$ 0 ≤ s ≤ 1 , define $$E_s(A):=\{\theta \in [0,1]: \dim (\rho _\theta (A))<s\}$$ E s ( A ) : = { θ ∈ [ 0 , 1 ] : dim ( ρ θ ( A ) ) < s } . We have $$\dim (E_s(A))\le \max \{0,1+\frac{s-\dim (A)}{2}\}$$ dim ( E s ( A ) ) ≤ max { 0 , 1 + s - dim ( A ) 2 } .

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“…which consists of directions to which the projection of A has zero measure. We notice that recently Harris [3]…”

Section: Remarkmentioning

confidence: 98%