Quantitative visibility estimates for unrectifiable sets in the plane

TL;DR

This paper investigates the visibility of unrectifiable sets in the plane, providing quantitative bounds on how these sets and their neighborhoods are perceived from points in the plane, extending previous qualitative results.

Contribution

It introduces explicit upper bounds for the visibility of neighborhoods of unrectifiable self-similar sets and establishes lower bounds for more general sets using advanced sum-product estimates.

Findings

Upper bounds on visibility of neighborhoods of unrectifiable sets.

Lower bounds on visibility for general sets.

Application of Bourgain's sum-product estimates.

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