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<p>For each <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k\ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we determine the dimensional threshold for planar fractal percolation to contain <i>k</i> 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.</p>

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The largest slice of fractal percolation

  • 期刊论文

  • 发表日期：

    2025年11月5日

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  • Pablo Shmerkin,
  • Ville Suomala

摘要

For each \(k\ge 3\) k ≥ 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.