### Abstract

Let A be an expanding integer n×n matrix and D be a finite subset of Z^{n}. The self-affine set T=T(A,D) is the unique compact set satisfying the equality A(T)=∪_{dD}(T+d). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈Z^{n}, and the measure of the intersection of self-affine sets T(A,D_{1})∩T(A,D_{2}) for different sets D_{1}, D_{2}⊂Z^{n}.

Original language | English (US) |
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Pages (from-to) | 389-393 |

Number of pages | 5 |

Journal | Discrete and Computational Geometry |

Volume | 46 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1 2011 |

### Keywords

- Graph-directed system
- Self-affine set
- Self-similar action
- Tile

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Bondarenko, I. V., & Kravchenko, R. V. (2011). On Lebesgue Measure of Integral Self-Affine Sets.

*Discrete and Computational Geometry*,*46*(2), 389-393. https://doi.org/10.1007/s00454-010-9306-8