Quantitative visibility estimates for unrectifiable sets in the plane

Abstract

The "visibility" of a planar set $S$ from a point $a$ is defined as the normalized size of the radial projection of $S$ from $a$ to the unit circle centered at $a$. Simon and Solomyak (Real Anal. Exchange 2006/07) proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of $\delta$-neighbourhoods of such sets. We also prove lower bounds on the visibility of $\delta$-neighborhoods of more general sets, based in part on Bourgain's discretized sum-product estimates

42 pages, 3 figures. v2: comprehensive revision, following detailed suggestions by a referee

References in corpus (3)

• On self-similar sets with overlaps and inverse theorems for entropy
• On planar self-similar sets with a dense set of rotations
• An estimate from below for the Buffon needle probability of the four-corner Cantor set

Cited by in corpus (7)

• On the dimension and smoothness of radial projections
• A sharp exceptional set estimate for visibility
• Multiplication on uniform $\lambda$-Cantor sets
• Transversal families of nonlinear projections and generalizations of Favard length
• Visibility of Cartesian products of Cantor sets
• On Hausdorff dimension of radial projections
• Quantitative Besicovitch projection theorem for irregular sets of directions