The largest slice of fractal percolation

TL;DR

The paper determines exact thresholds for the appearance of $k$ collinear points in dyadic planar fractal percolation, showing a sharp transition at $p_k=2^{(-k-2)/k}$ for $k\ge3$ and that no $k$-point line exists at criticality. It combines discretization, first- and second-moment methods, and FKG-type arguments to prove nonexistence below the threshold and existence above it (conditioned on non-extinction), establishing a cascade of phase transitions as $k$ grows. In the critical case where the Hausdorff dimension is $1$, the largest linear slice is a Cantor-type set with zero Hausdorff dimension in the standard sense, but it attains positive size in refined gauge-based measures: for a suitable $\phi$, there exists a line with $\mathcal H^\phi(A\cap\ell)>0$, and $\mathcal H^\phi(A)<\infty$. The results extend to $m$-ary grids with $p_k=m^{(-k-2)/k}$ and the same dimension-threshold formula, highlighting a deep link between fractal percolation geometry and generalized Hausdorff measures.

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.

Figures (2)

• Figure 1: A realization of fractal percolation with $p=0.8$.
• Figure 2: A possible selection of $R_i$ for $k=3$ and $m=2$. The unions of the squares of the same colour form two elements of $\mathcal{C}_2$.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Proposition 3.1
• proof : Proof of Theorem \ref{['thm:finite-intersections']} assuming Proposition \ref{['prop:discretized_statement']}