Upper and lower bounds on the rate of decay of the Favard curve length for the four-corner Cantor set

2022

J Geom Anal

The Besicovitch projection theorem states that if a subset E of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of E to a line will have zero measure. In other words, the Favard length of a purely unrectifiable 1-set vanishes. In this article, we show that when linear projections are replaced by certain non-linear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch non-linear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its Favard curve length is very small. Our techniques build on those of Tao, who in (Proc Lond Math Soc 98:559-584, 2009) proves a quantification of the original Besicovitch projection theorem.

Section: Introduction and Main Resultsmentioning

confidence: 93%

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Quantification of a Besicovitch Non-linear Projection Theorem via Multiscale Analysis

Davey

2022

J Geom Anal

“…We note that a deeper investigation of the result of Corollary 2.6 is followed up on in [2, 3], where we obtain upper and lower bounds on the rate of decay as

n\rightarrow \infty

of the probability that a unit circle intersects the n th generation in the construction of the four corner Cantor set,

C(1/4)

. Further, in [2, 4], we obtain similar quantitative information for more general irregular 1‐sets.…”

Section: Main Results When γ Has Non‐vanishing Curvaturementioning

confidence: 99%

“…If we apply Theorem 2.3 for (1∕𝑟) ⋅ 𝐴 instead of 𝐴, then we obtain that 3 is open (since we assumed that 𝐴 is compact) so, Θ is closed. By (2.3) and the Fubini Theorem, we know that  3 (Θ) = 0.…”

Section: Pinned Distance Setsmentioning

confidence: 99%

See 1 more Smart Citation

Dimension and measure of sums of planar sets and curves

Simon

2022

Mathematika

Self Cite

Considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the measure and dimension of 𝐴 + Γ ∶= {𝑎 + 𝑣 ∶ 𝑎 ∈ 𝐴, 𝑣 ∈ Γ} when 𝐴 ⊂ ℝ 2 and Γ is a piecewise  2 curve. Assuming Γ has non-vanishing curvature, we verify that:if and only if 𝐴 is an irregular (purely unrectifiable) 1-set.In this article, we develop an approach using nonlinear projection theory which gives new proofs of (a) and (b) and the first proof of (c). Item (c) has a number of consequences: if a circle is thrown randomly on the plane, it will almost surely not intersect the four corner Cantor set. Moreover, the pinned distance set of an irregular 1-set has 1-dimensional Lebesgue measure equal to zero at almost every pin 𝑡 ∈ ℝ 2 .M S C 2 0 2 0 28A75 (primary), 28A80, 42B10 (secondary)

Transversal families of nonlinear projections and generalizations of Favard length

Bongers

2023

Analysis & PDE

Self Cite

Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a transversality condition. We establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four-corner Cantor set, first established by Cladek, Davey, and Taylor.