The largest slice of fractal percolation

TL;DR

This paper determines the dimensional threshold for planar fractal percolation to contain multiple collinear points and analyzes the size of the largest linear slice in the critical case, revealing it as a zero-dimensional Cantor set.

Contribution

It establishes the precise dimension thresholds for collinearity in fractal percolation and characterizes the size of the largest linear slice at the critical dimension.

Findings

Thresholds for collinear points in fractal percolation are identified.

The largest linear slice at critical dimension is a zero-dimensional Cantor set.

Generalized Hausdorff measures quantify the size of the linear slice.

Abstract

For each $k\ge 3$, we determine the dimensional threshold for planar fractal percolation to contain $k$ collinear points. In the critical case of dimension $1$, the largest linear slice of fractal percolation is a Cantor set of zero Hausdorff dimension. We investigate its size in terms of generalized Hausdorff measures.