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Let \(F\subseteq [0,1]^2\) be a Bedford-McMullen carpet defined by exponents \(m>n\) , that projects to [0, 1] on the y-axis. We show that under mild conditions on F, there are many non principle lines \(\ell \) such that \(\dim ^* F\cap \ell = \dim ^* F -1\) , where \(\dim ^*\) is Furstenberg’s star dimension (maximal dimension of a microset). This exhibits the sharpness of recent Furstenberg-type slicing theorems obtained by Algom (2020) about upper bounds on the dimension of every such slice.

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Large Slices Through Self Affine Carpets

  • 会议论文

  • 发表日期：

    2025年

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  • Amir Algom,
  • Meng Wu

摘要

Let \(F\subseteq [0,1]^2\) be a Bedford-McMullen carpet defined by exponents \(m>n\) , that projects to [0, 1] on the y-axis. We show that under mild conditions on F, there are many non principle lines \(\ell \) such that \(\dim ^* F\cap \ell = \dim ^* F -1\) , where \(\dim ^*\) is Furstenberg’s star dimension (maximal dimension of a microset). This exhibits the sharpness of recent Furstenberg-type slicing theorems obtained by Algom (2020) about upper bounds on the dimension of every such slice.