### Abstract

For Lebesgue typical (θ, a), the intersection of the Sierpinski carpet F with a line y = x tan θ + a has (if non-empty) dimension s - 1, where s = log8/ log 3 = dim _{H} F. Fix the slope tan θ ∈ ℚ. Then we shall show on the one hand that this dimension is strictly less than s - 1 for Lebesgue almost every a. On the other hand, for almost every a according to the angle θ-projection ν ^{θ} of the natural measure ν on F, this dimension is at least s-1. For any θ we find a connection between the box dimension of this intersection and the local dimension of ν ^{θ} at a.

Original language | English |
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Pages (from-to) | 213-250 |

Number of pages | 38 |

Journal | Transactions of the American Mathematical Society |

Volume | 365 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 20 2012 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Manning, A., & Simon, K. (2012). Dimension of slices through the sierpinski carpet.

*Transactions of the American Mathematical Society*,*365*(1), 213-250. https://doi.org/10.1090/S0002-9947-2012-05586-3