The visible part of plane self-similar sets

TL;DR

This paper proves that for a broad class of self-similar fractal sets in the plane with Hausdorff dimension at least 1, the visible part from almost every direction has Hausdorff dimension 1, confirming a conjecture under certain conditions.

Contribution

It establishes that self-similar sets satisfying the convex open set condition and with interval projections have visible parts of Hausdorff dimension 1 from almost all directions.

Findings

Visible parts have Hausdorff dimension 1 for almost all directions.

The result applies to self-similar sets with rotations and possibly disconnected.

Confirms a conjecture for a broad class of fractal sets.

Abstract

Given a compact subset $F$ of $\mathbb{R}^2$, the visible part $V_\theta F$ of $F$ from direction $\theta$ is the set of $x$ in $F$ such that the half-line from $x$ in direction $\theta$ intersects $F$ only at $x$. It is suggested that if $\dim_H F \geq 1$ then $\dim_H V_\theta F = 1$ for almost all $\theta$, where $\dim_H$ denotes Hausdorff dimension. We confirm this when $F$ is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of $F$ onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and $F$ need not be connected.