### 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 |
---|---|

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 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*365*(1), 213-250. https://doi.org/10.1090/S0002-9947-2012-05586-3

**Dimension of slices through the sierpinski carpet.** / Manning, Anthony; Simon, K.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 365, no. 1, pp. 213-250. https://doi.org/10.1090/S0002-9947-2012-05586-3

}

TY - JOUR

T1 - Dimension of slices through the sierpinski carpet

AU - Manning, Anthony

AU - Simon, K.

PY - 2012/8/20

Y1 - 2012/8/20

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84861360478&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861360478&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2012-05586-3

DO - 10.1090/S0002-9947-2012-05586-3

M3 - Article

VL - 365

SP - 213

EP - 250

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -